Hello there.
That's a very good question you have there. I can only speak to Engineering - but I know that Columbia provides housing to incoming engineering PhD's.. Through one of their portals, you can indicate your preferred price range and then they will evaluate your profile. Current grad students or advisors/administrators are very helpful and will be able to offer you good advice as well.
NYC is great because of its metro / bus / taxi system - you can get pretty much anywhere relatively quickly. And, of course, prices can vary greatly between the five Burroughs.
I applied to some of the schools mentioned above - I am an international student.
Haven't heard anything yet from these schools. Hopefully, the results start coming in shortly.