Tenderloin/Mission/Sunset are places you might be able to find something for a grand.

Tenderloin - stabby stabby. Mission - on the dirtier side, a little sketchy in parts, but trendy and experiencing lots of gentrification. Sunset - lots of asian influence, very livable. You'll very rarely see the sunset from here though.

You might even be able to find something in Richmond for a grand if you're lucky. Just keep an eye out, set up alerts on padmapper.com. I found a $1400 one bedroom in a nice part of the city there.