Why travel from New York to California by ship, in the 1860s?

Because it was faster and much safer.

Traveling by coach wasn't a thing. Much more likely would be to go with some pioneer wagons. We know that it took them around 4-5 months, that is between 120-170 days (depending on where in California).

In 1861, the Pony Express (or lone horse riders) claimed that it would take 10 days to deliver letters from New York to Sacramento. The fine print was that it was actually 10 days to deliver from Saint Joseph, Missouri to Sacramento, since the message was sent there by telegraph.

And finally by boat - it took 43 days if you chose the Isthmus of Panama route. Around 100 - 200 days if, as in the movie, you took the long route around Cape Horn. In 1853 the clipper Flying Cloud made the Cape Horn route in record time of 89 days.

And finally, price. The ship route was much cheaper than any other one. Ships have usually taken goods and cargo, so passengers were a nice additional profit. On a wagon trail you had to be useful, you had to pay for your own food, and you would need to travel to Missouri to start the journey. So it's just easier to go straight from New York by ship.

The movie takes place in 1862 (according to Wikipedia), and the transcontinental railroad was started in 1863, so the alternative is something like the Oregon trail. Jenny isn't a pioneer, just a traveler. On a boat, things like food and shelter would be mostly taken care of.

Also, I suspect it's a bit more romantic, so makes for a better movie.

Short Answer:

One big advantage is that someone could buy a ticket to California in New York City for a voyage on a ship leaving New York City for California. The passenger would not have to travel a thousand miles by railroad to Missouri to get a stagecoach or join a wagon train, and they would not have to make detailed plans and buy equipment for a wagon train journey.

Long Answer:

Travelling from New York to California before the transcontinental railroad opened in 1869 would be long, expensive, and dangerous. But going by ship would probably be the best way to go, and it would certainly be the simplest to plan. Just buy a ticket on a ship leaving from the busy port of New York City. There would be no need to find out what to buy for a wagon train trip or buy a train ticket to a terminal of a stage line and then buy a stage ticket.

Most people who moved to California overland joined a wagon train in western Missouri,

The California Trail was an emigrant trail of about 3,000 mi (4,800 km) across the western half of the North American continent from Missouri River towns to what is now the state of California. After it was established, the first half of the California Trail followed the same corridor of networked river valley trails as the Oregon Trail and the Mormon Trail, namely the valleys of the Platte, North Platte, and Sweetwater rivers to Wyoming. In the present states of Wyoming, Idaho, and Utah, the California and Oregon trails split into several different trails or cutoffs.

https://en.wikipedia.org/wiki/California_Trail1

As one can guess from the distance traveled, it took months to reach California.

The cost of traveling over the California or Oregon trail and its extensions varied from nothing to a few hundred dollars per person. Women seldom went alone outside of family groups and were a distinct minority in the West for decades. The cheapest way to travel the trail was to hire on to help drive the wagons or herds, allowing one to make the trip for nearly nothing or even make a small profit. Those with capital could often buy livestock in the Midwest and drive the stock to California or Oregon and usually make good money doing it. About 60–80% of the travelers were farmers, and as such already owned a wagon, livestock team and many of the necessary supplies, this lowered the cost of the trip to about $50.00 per person for six months food and other items. Families often planned for a trip months in advance and made many of the extra clothing and other items needed. Individuals buying most of the needed items would end up spending between $150 and $300 per person.[143] Some who traveled in "grand" style with several wagons and servants could spend much more.

As the trail matured, additional costs for ferries and toll roads were thought to have been about $30.00 per wagon or about $10.00/person.[144]

https://en.wikipedia.org/wiki/California_Trail#Costs2

The route West was arduous and filled with many dangers, but the number of deaths on the trail is not known with any precision, and there are only wildly varying estimates. The estimates are made even harder by the common practice then of burying people in unmarked graves that were intentionally disguised to avoid them being dug up by animals or Indians. Graves were often put in the middle of a trail and then run over by their livestock to make them difficult to find. Diseases like cholera were the main killer of trail travelers with up to 3% (or more) of all travelers (6,000 to 12,000+ total) dying of cholera in the cholera years of 1849 to 1855. Indian attacks were probably the second leading cause of death with about 500 to 1,000 being killed from 1841 to 1870. Other common causes of death included: freezing to death (300–500), drowning in river crossings (200–500), getting run over by wagons (200–500), and accidental gun deaths (200–500).

https://en.wikipedia.org/wiki/California_Trail#Deaths3

There was another way to get to California overland, by using the Santa Fe trail from Missouri to Santa Fe, New mexico, then travelling south along the Rio Grande to near the Mexican border, and heading west to the Gila River and following it to the Colorado River and crossing into California. That route went through lands of the dangerous Comanche and Apache tribes and hundreds of miles of Southwestern deserts.

Other routes to California included sailing to Mexico, Nicaragua, or Panama, crossing overland to the Pacific Ocean, and sailing to California.

In what has been referred to as the "first world-class gold rush,"[14] there was no easy way to get to California; forty-niners faced hardship and often death on the way. At first, most Argonauts, as they were also known, traveled by sea. From the East Coast, a sailing voyage around the tip of South America would take four to five months,[15] and cover approximately 18,000 nautical miles (21,000 mi; 33,000 km). An alternative was to sail to the Atlantic side of the Isthmus of Panama, take canoes and mules for a week through the jungle, and then on the Pacific side, wait for a ship sailing for San Francisco.[16] There was also a route across Mexico starting at Veracruz. The companies providing such transportation created vast wealth among their owners and included the U.S. Mail Steamship Company, the federally subsidized Pacific Mail Steamship Company, and the Accessory Transit Company. Many gold-seekers took the overland route across the continental United States, particularly along the California Trail.[17] Each of these routes had its own deadly hazards, from shipwreck to typhoid fever and cholera.[18] In the early years of the rush, much of the population growth in the San Francisco area was due to steamship travel from New York City through overland portages in Nicaragua and Panama and then back up by steamship to San Francisco.[19]

https://en.wikipedia.org/wiki/California_Gold_Rush#Transportation_to_California4

U.S. Mail Steamship Company was a company formed in 1848 by George Law, Marshall O. Roberts and Bowes R. McIlvaine to assume the contract to carry the U. S. mails from New York City, with stops in New Orleans and Havana, to the Isthmus of Panama for delivery in California. The company had the SS Ohio and the SS Georgia built in 1848, and with the purchased SS Falcon in early 1849 carried the first passengers by steamship to Chagres, on the east coast of the Isthmus of Panama. Soon the rapid transit time the steamship lines and the trans isthumus passage made possible when the California Gold Rush began made it a very profitable company.

https://en.wikipedia.org/wiki/U.S._Mail_Steamship_Company5

The Accessory Transit Company was a company set up by Cornelius Vanderbilt and others during the California Gold Rush in the 1850s, to transport would-be prospectors from the east coast of the United States to the west coast.

At the time, an overland journey across the US was an arduous undertaking and could last many weeks. The Accessory Transit Company instead took passengers by steamer from New York to San Juan del Norte on the Caribbean or Mosquito Coast of Nicaragua. From there, they travelled up the Rio San Juan to Lake Nicaragua, crossing the lake to the town of Rivas. A stagecoach then crossed the narrow isthmus to San Juan del Sur, where another steamer travelled to San Francisco.

The ATC provided the cheapest route to California from the east coast, and was soon carrying 2,000 passengers a month at a fare of $300 each, later reduced to $150.

in 1857 Vanderbilt discontinued this route in return for monthly payments from the competing companies.

https://en.wikipedia.org/wiki/Accessory_Transit_Company6

Buying passage on a ship sailing around South America by Cape Horn increased the travel time and increased the traveler's exposure to the dangers of sea travel, but totally avoided all the discomfort and danger of the California Trail or traveling though the disease ridden jungles of central America.

So choosing to take the longer sea route was a fairly reasonable choice considering that there wasn't any really good way to get to California before the transcontinental railroad was completed in 1869.

Another advantage is that someone could buy a ticket to California in New York City for someone to take a ship leaving New York City for California. The passenger would not have to travel a thousand miles by railroad to Missouri to get a stagecoach or join a wagon train, and they would not have to make detailed plans and buy equipment for a wagon train journey.

Added April 9, 2020. I see that I forgot to mention passenger stage coach routes to California.

Butterfield Overland Mail (officially the Overland Mail Company)1 was a stagecoach service in the United States operating from 1858 to 1861. It carried passengers and U.S. Mail from two eastern termini, Memphis, Tennessee, and St. Louis, Missouri, to San Francisco, California. The routes from each eastern terminus met at Fort Smith, Arkansas, and then continued through Indian Territory (Oklahoma), Texas, New Mexico, Arizona, Mexico, and California ending in San Francisco.2 On March 3, 1857, Congress authorized the U.S. postmaster general, Aaron Brown, to contract for delivery of the U.S. mail from Saint Louis to San Francisco.3 Prior to this, U.S. Mail bound for the Far West had been delivered by the San Antonio and San Diego Mail Line (Jackass Mail) since June 1857.4

The Butterfield stages may have been the first passenger stages to California. Their trail through the Southwest was called the Southern Overland Trail.

The Central Overland Route (also known as the "Central Overland Trail", "Central Route", "Simpson's Route", or the "Egan Trail") was a transportation route from Salt Lake City, Utah south of the Great Salt Lake through the mountains of central Nevada to Carson City, Nevada. For a decade after 1859, until the first Transcontinental Railroad was completed in 1869, it served a vital role in the transport of emigrants, mail, freight, and passengers between California, Nevada, and Utah.

https://en.wikipedia.org/wiki/Central_Overland_Route7

In 1861 John Butterfield, who since 1858 had been using the Butterfield Overland Mail route through the deserts of the American Southwest, also switched to the Central Route to avoid possible hostilities during the American Civil War. The various stage lines, by traveling day and night and changing their teams at about 10 miles (16 km) to 20 miles (32 km) intervals, could get light freight, passengers, and mail to or from Missouri River towns to California in about 25–28 days.

https://en.wikipedia.org/wiki/Central_Overland_Route#Improvement8

In 1869 the First Transcontinental Railroad was completed using the more level route along the Humboldt River to the north—along much of the original California Trail route. Passengers and freight were now carried faster and cheaper than on the stage and freight lines using the old route.

https://en.wikipedia.org/wiki/Central_Overland_Route#Obsolescence9