If you look up New York budget accommodation on the Internet you will find
lots of places offering such accommodation. I stayed in such a place 3 years
ago - aimed at students, backpackers etc.
I always obtain a Youth Hostel card when I travel just in case I need to
stay in one. These days you don't generally need to be a member beforehand.
I stayed in the one in Philadelphia which was in a lovely historic house.
Naturally the main thing is to make sure that you and your belongings are