Fort Greene rents went through the roof about 3 years ago, but they might have calmed down a little. You probably won't be able to find anything decent for under $1600, though. Clinton Hill is a little better, but also a little further out, and can be sketchy sometimes - our old upstairs neighbors moved there and tell us about shit like young teenagers throwing stuff at their 6-year-old son, or open gang brawls in the street. It does vary a lot block by block, though.
In terms of proximity to Manhattan, though, subway access matters a lot more than how close you are on the map (within reason; Coney Island or Rockaway are going to take forever to get you into Midtown, let alone 115th street). If you must live in Brooklyn, try to get something off of the A (Boerum Hill probably has some good deals, especially by the Hoyt-Schermerhorn stop) or the 2/3 (Prospect Heights and Crown Heights come to mind, though both are fairly far out).