Why is Supergirl's earth numbered 38?

Most probably the number thirty eight was chosen to mark the debut of Superman in the year of 1938.

Superman debuted this season in Arrowverse. So, to associate it with the debut of the Golden age Superman, Earth 38 was chosen.

Also, in this very Earth 38, Supergirl takes the name of Kara in her civilian life (nothing new and non-typical).

As Wikia writes at the outsets:

Home to a Superman that debuted in 1938 and a Batman that first appeared in 1939. ...

And its continuation in Earth-3839:

Earth-3839 is a reality in which Superman and Batman started their careers in the 1930s, and started families that would follow in their superhero footsteps all the way to the 30th Century.

Also, summing up the post with the CINEMA BLEND article:

[...] I'm not sure how the folks at The CW chose 38 as the number for the Supergirl universe, but it could be a nod to the fact Superman made his comic book debut in 1938 as the first major superhero. ... Supergirl wouldn't exist without Superman, so Earth-38 could be a solid way to honor the superhero who started it all.

Albeit it's a speculation, I deem it as a reasonable explanation as to why they chose 38.

The 38 is, apparently, a reference/homage to 1938, the year in which the first Superman comic was published.

Wikia - As far as the comics are concerned:

Earth 38 is the Earth of the 'local' New 52 Multiverse most accurately reflecting DC's publishing history, as heroes appear on this world when they did in the real world. It is therefore probable that comics throughout the local 52-Earth Multiverse featuring 'traditional'-looking versions of Superman, such as the DC Comics of Earth 5 and Earth 10 (Earth 10 in particular publishing Superman in 1938, this being the only Superman of the Multiverse known to operate in that timeframe), were in fact representations of the Superman of Earth 38.

and

Den of Geek.com

According to Grant Morrison's guide to the DC Multiverse, Multiversity, Earth-38 is the world where John Byrne's series of minis called Generations takes place. What the hell is Generations? Well, the short version is that imagine a world where Superman first appeared in 1938 (when his comics first started getting published), and Batman followed in 1939 (again, with the comics). Now, on this world, they both aged and developed in real time, had kids, etc, and these stories follow them and their offspring through the present day and beyond.

I honestly thought it was simply corresponding to the season and episode of "The Flash" which mentioned it (which was Season 3, Episode 8, or, more specifically...3-8).

When they do these as easter eggs, it can be for many coincidental reasons. The superman 1938 analysis is fine, but here's another idea: Melissa Benoist is from Colorado (her home state.) Colorado is the 38th state in the US. Not sure if that played into the decision, but it's a heck of a coincidence.