We went through this hell last summer. Here's some tips that we used to stay away from brokers (which is near impossible).
1. Apartments.com has all the major non-fee big high-rises. They are luxury and you will pay a premium, but some are reasonable.
2. Apartmentratings.com has good reviews on the worst buildings. If the building is not on there, it's probably OK as most of the ratings are just complaints.
3. Listingsquare.com is a REALLY good resource. We had a broker for awhile and we realized he was just going to this site and showing us all those apartments (thus getting the whole 10% for himself). We dropped him.
4. Archstoneapartments.com is where we ended up. It's a good place, clean a little section 8 problems (about 20% is very low income) but the place is brand new and lots of amenities. More importantly, they only require ONE months rent, no deposit, no last month, no fee. If you want a referral, PM me and I'll split whatever they give me.
5. Move to outlying parts of NYC - Brooklyn/Queens/Jersey City/Hoboken. There's more deals out there.
6. Remember, in the city you probably need to make 40 x 1 months rent yearly to qualify. Most of the time you can combine two incomes or use a co-signer.
Good luck