I'm a little more optimistic about the cost of living in the city. I've been here for 6 years, and I live in a studio on the Upper East Side of Manhattan (I've been in this apt for a couple of years) in a safe neighborhood right near the subway for just over $1k. This is an unusually good deal, sure, but I found it on Craigslist so it can be done. Friends of mine have shares in the "good" parts of Brooklyn for $700-$900. It can definitely be done. Another friend lives in a beautiful one bedroom with a modern kitchen and walk-in closet in Carroll Gardens (a lovely, safe, convenient part of Brooklyn) that she found through a broker recently for $1400. (Not that you could necessarily afford that, but just to give an idea.)
Apartment prices have definitely gotten cheaper over the past 6 months.
I can't really say about living on a certain amount with a child, since I don't have one, but I think if you budget carefully, and aren't making large payments on other debts, etc. a single person can definitely live in NYC on that amount.
Good luck!