In Fantastic Beasts and Where to Find Them, Newt is introduced first as we see him go through immigration, presumably on Ellis Island.
In the very next scene featuring Newt, we see him apparate through a solid wall, into the back room of a bank. This means that:
- Newt is a qualified apparator, he knows how to move from one place to another by magic.
- Newt is happy to apparate to somwhere he can't see and he's never been before
- We also see him apparate with his case, so that's clearly safe, too.
In the scope of the series we see wizards apparate between continents in the blink of an eye. It seems it's no more difficult than travelling across the room.
So why does he take a steam ship to New York? Only to risk being found out as he spends time with muggles on board, and going through the immigration process?