In general, a bank won't consider you for a mortgage with less than 20% down (unless you're willing to pay for private mortgage insurance, which is not a good road to go down), and most co-ops allow won't allow more than 80% financing, so what you have in the bank would need to be equal to 20% of the price of the home (+ another 2-3% for closing costs).
That said, money in the bank isn't the only thing considered when banks are evaluating a loan, and there are programs out there for folks with not a lot of $ to put down. If you have a good credit score and little to no debt, you could be an attractive borrower regardless of how much $ is currently in the bank. And given your income situation, you likely qualify for any number of programs that exist for first-time homebuyer assistance. See these links:
http://www.nyc.gov/html/hpd/html/buyers/downpayment.shtmlhttp://www.nyshcr.org/Topics/Home/Buyers/There are additional programs out there if you are a military vet.
All that said, NYC co-ops tend to have stricter financial requirements than lenders, so even with assistance you may still have difficulty. You could look at condos, which cost more up front but tend to not scrutinize finances as heavily.
Another option is to look for city sponsored affordable condos/homes:
http://www.nyc.gov/html/hpd/html/buyers/lotteries.shtmlAgain, given your situation, you seem to qualify for this program, and if the Mayor's new housing plan is pushed forward, you could see a LOT of affordable condos popping up around town (perhaps not in JH, but certainly other parts of Queens). So there's some waiting involved, as well as some luck (if there are more qualified purchasers than units, there is usually a lottery), but something to put on your radar.
I'm not a realtor, lender, or government employee with any expert knowledge of any of the above, just someone who, like you, is wanting to purchase a home and see what options are out there. The above is a bit of what I've come across in my research so far.
Good luck!