We don't have "backpackers/hostels" in the US.
Personally, I'd avoid NYC at New Year's. It's mostly an unmanageable mob of crowds wandering around pissing and vomiting. Around midnight you can't even walk, it's like sardines. Think Mardi Gras but without the party vibe and an NYC angry attitude. That was my NYC New Year experience.
Places I've had a great time, not so far from there, on New Year's were Montreal and Boston. Far lesser crowds and a little more common courtesy from people before they toss cheap beer on you.
I'm also a native New Yorker and I'd say avoid Times Square on New Year's Eve.
I don't know anyone who's been to Times Square on New Year's Eve more that once. It's one of those things you do to experience it, and then wonder why it's so popular. It's too crowded, you are too far away to see anything and people get rowdy.
Just to get things straight, New Yorker's are very nice people and are usually eager to help tourist's find their way around. Any rowdiness you'd find on New Year's Eve is an anomally (and remember that most of the ball gawkers on Dec. 31st are also tourists). NYC is not an angry city by any means.
I say come to New York for New Year's but make plans or reservations away from Time's Square on the 31st. There are many great places to go (even outside Manhattan) where you won't be crushed by crowds. Oh, and stay for a least three days.

I'm not entirely sure of any hostels around NYC (and I'd probably avoid those anyhow), but it pays to do some research on travel websites to see if something similar is available. And you don't
have to sleep in Manhattan - the city is a quick, cheap train ride away, even from Long Island - where motels are more affordable, comfortable and clean.