Absolutely.
SInce finances are a concern, you’ll want to look at either SUNY or CUNY (State University/ City University of NY). Even as an out of state applicant, you’ll probably do better financially than you would with a private school.
Here’s an old thread on the issue: http://talk.collegeconfidential.com/visual-arts-film-majors/1245204-best-cuny-or-suny-for-art.html
Just be sure, of course, that your choice has dorms. http://www.collegesimply.com/colleges/new-york/colleges-with-dorms/
. And beware when you look at SUNY-- you’ll NEED to see where in NY they are if you want to be in or near NYC. Many are 8-10 hours away from NYC.
Also, take a look at schools in NJ-- some are just a few minutes by train from NYC.