Good areas to live in New York?

Totally delayed reply, but seems it's not too late.

Greenpoint YMCA is well located and very cheap (because the Y doesn't have to pay hotel tax) and you get your own room although the bathroom facilities are shared.

Pros: Good location, swimming pool and sauna in basement, cheap as chips, authentic experience (lots of the residents are just that).

Cons: Lots of the residents are just that, opposite the fire station so can be noisy at night, mice, not exactly romantic.

Did I say it's cheap though?

Does anyone know of a cheap place to stay in New York? Wanna take my girlfriend there for a surprise break next september, but need to do it on quite a tight budget. Hostels don't really appeal as I would like some privacy.