Quite honestly I would not stay at any place in Nyc if you don't know them before hand.
Asking to crash at someone place in Ny from a random stranger is just asking for it, especially in Nyc. Trust me I'm from Nyc and there are a lot of scum bags in the city. There are good ones but usually the good ones will not let random strangers stay in their apartment or house.
You can most likely find a cheap hotel if you are in upstate NY. Nothing in Nyc is cheap.