I've just come across this explanation in the book "Quidditch Through the Ages" .. page 90:

"Too cumbersome to turn at high speed, the Oakshaft [79] never gained much popularity with those who prized agility over safety, though it will always be remembered as the broom used in the first ever Atlantic broom crossing, by Jocunda Skyes in 1935. (Before that time, wizards preferred to take ships rather than trust broomsticks over such distances. Apparation becomes increasingly unreliable over very long distances, and only highly skilled wizards are wise to attempt it across continents.)" - Quidditch Through the Ages

Looking at this, since Newt went to America in 1920s, it would be safe to assume he did not use other means of transport for this reason.

Similarly, portkeys are unpleasant to travel.. cause nausea and giddiness making it not exactly travel-friendly all the time.

The next reason is that portkeys between continents need to be approved by both the ministries and these are monitored. When Newt is bringing a suitcase full of beasts, I think he would try and maintain a low key with MACUSA (which has quite strict rules and regulations) - HP Wikia

Thanks to Rand al'Thor, here's a run-down of modes of transport that we know of that could and couldn't work as international modes of transport:

Portkeys - yes.

The answers to this question show that Portkey travel can be international. Copying quotes from my and DVK's answers over there:

• from various places to the Sahara desert:

"While there have been no proven referee slayings since [1357], there have been several incidences of broom-tampering over the centuries, the most dangerous being the transformation of the referee's broom into a Portkey, so that he or she is whisked away from the match halfway through and turns up months later in the Sahara Desert."

_{-- Kennilworthy Whisp, Quidditch Through the Ages}

• from France to the Burrow:

Mr. Weasley had therefore gone to meet the Delacours on top of a nearby hill, where they were to arrive by Portkey

_{-- HP and the Deathly Hallows, Chapter 6: The Ghoul In Pyjamas}

• from Godric's Hollow to eastern Europe:

“It was poor little Ariana dying, I think, that did it,” says Bathilda. “It came as an awful shock. Gellert was there in the house when it happened, and he came back to my house all of a dither, told me he wanted to go home the next day. Terribly distressed, you know. So I arranged a Portkey and that was the last I saw of him

_{HP and the Deathly Hallows, Chapter 19, The Life and Lies of Albus Dumbledore}

Apparition - usually no.

The answers to this question show that there is some confusion and contradiction over whether or not Apparition can be international. Firstly, as stated in KeithS's answer, this quote from the books tells us that there is a limit on the distance even a skilled wizard can Apparate:

Harry knew it; his scar was bursting with the pain of it, and he could feel Voldemort flying through the sky from far away, over a dark and stormy sea, and soon he would be close enough to Apparate to them, and Harry could see no way out.

_{-- HP and the Deathly Hallows, Chapter 23: Malfoy Manor}

The following quote from Kevin's answer tells us that it is possible but risky to Apparate across continents:

Apparition becomes increasingly unreliable over very long distances, and only highly skilled wizards are wise to attempt it across continents.

_{-- Quidditch Through the Ages, Chapter 9}

And finally, Richard's answer tells us that Apparition between countries is banned.

Floo - unclear.

All the information we have in canon concerns the Floo Network within Britain. This includes the latest articles from Pottermore by JK Rowling. Quoting from the second of these sources:

Floo powder was invented by Ignatia Wildsmith in the thirteenth century. Its manufacture is strictly controlled. The only licensed producer in Britain is Floo-Pow, a company whose Headquarters is in Diagon Alley, and who never answer their front door.

The inclusion of the words "in Britain" would seem to suggest that there are Floo Networks in other countries too, but doesn't tell us whether they're connected.

Why did Newt use a normal, muggl—NoMaj's mode of transport

We know from the movie Fantastic Beasts and Where to Find Them that Newt didn't want to be detected by the MACUSA, and we know that most magical modes of transport (including apparition and floo transport) can be traced, detected or otherwise tampered with. From this we can infer that Newt used a conventional NoMaj form of transport to decrease the likelihood of alerting the authorities.

An out-of-universe explanation might be that it is a dramatically boring way to introduce us, the audience, to Newt's arrival to America, compared to the gorgeous, The Godfather II-esque imagery of the immigrant ship. It also serves, dramatically, to tell us, the audience, that he is a foreigner in the US, much like we are alien to a magical 1920's America set in the Potterverse, and the movie can achieve this introduction without a single word of exposition to explain it.

^{_{Note: A lot of this is speculation until confirmation starts inevitably pouring in from recent interviews with JK Rowling (and the rest of production team).}}

@ibid on scifi.SE found a JKR quote to support an answer about the max range of portkey travel which exactly answers this question, too:

From this Twitter exchange:

Q: Loved the movie. Burning Q from my wife though: How come Newt travels by boat to cross the Atlantic instead of apparating?

J.K. Rowling: There's a limit to how far you can Apparate. Inter-continental Apparition is very dangerous!

Q: Why couldn't Newt have used a portkey? It can't have been difficult for him to register one since he works at the Ministry.

J.K. Rowling: Consider the effect on the contents of his case...

So yes a portkey could take you that far. But portkey travel's physical effects would be not just uncomfortable, but actually a problem for Newt's case and/or the contents.

I haven't seen the movie, so I don't know why. Either magical creatures are more sensitive, or it's like putting a bag of holding inside a portable hole, if it's a bigger-on-the-inside case.

So some of this may be slightly incorrect, but these are the reasons from what I understand:

Apparating

Newt couldn't apparate, because he had never been to where he needed to be before. Anyone who apparates in the HP universe goes somewhere that they have been to before, or that they can see directly.

Apparating randomly is very risky, as you can possibly appear inside of a wall or a tree, and as both wizard and static object can't occupy the same space simultaneously, it will cause a rather severe case of deadness. Even more so because it was over long distances, which causes it to be less stable anyway.

Portkeys

Portkeys need to be registered with the authorities, therefore it's very possible that if Newt had attempted to travel across continents he would have had to alert the Ministry of Magic, who would have informed MACUSA, and as Newt was illegally transporting magical creatures into the country, there was a reason he was doing it below the radar.

Whilst he did apply for a permit to travel, I'm assuming he traveled through muggle forms of transport so that he could go through muggle customs, as he may have needed to be subject to a search by wizard customs and they would have found out about the creature he was smuggling.

I doubt he could make his own unregistered portkey as, like with apparition, he would have no clue as to where to travel to.

Floo

I'm assuming that as the Floo Network is referred to as a 'network', this means they need to be connected to one another. I'm not sure if this network can span across continents, but even if they did the user needs to say exactly where they want to go when they travel. So Newt couldn't just say 'America' as he'd have no idea in which fireplace he would end up.

Perhaps he could set up somewhere to travel to, but he would need to trust that person with the reason for his visit, and considering the only person he mentions knowing in New York had his business shut down a year earlier and he didn't know about it, it's very possible that he had no one he could trust to keep his secret.

There could also be a simpler explanation that the magic to create Portkeys and Floo Networks wasn't invented before the 1920s, but I have no idea of the canonicity of that.

@Gray Roberts. Look at it this way Gray. Whether its from a No Mag or Muggle and Wizarding World stand point. For muggles in the 1920s the only means of transportation was either by car or boat, since airports were still rare and intercontinental transport by plane was still in the experimental stage. Since the Wright Brothers made their first flight in 1903, and Charles Lindbergh didn't make his first flight over the Atlantic from America to Germany in 1927. And America Airlines didn't start until 1930, with flights from America to Britain, Germany and other countries til the mid 30s. Besides the only means of transport after WWI by flight was by a Zeppelin. Which at that time only the rich could afford to fly or ride in a Zeppelin over the ocean. Which even then was also slow depending on the wind. So as time progress both in Muggle and Wizarding World technology advanced. So maybe possibly the ability to travel from England to America for witches and wizards weren't possible because maybe back then the wizard or witch who would invent the Port Key or Floo Powder didn't exist yet or were still in their infancy and experimental stages for long distance travel. For you cant teleport without splitching from time to time would make it risky or at worst the death of the witch or wizard. And as someone else said it could also be the ability for the America Wizard Ministry to keep track of witches and wizards who come to the country.