Holy crap do I think about this a lot. Here are my suggestions and the stuff I have put on when I've played in Apocalypse World games:
1) EARTH. Earth is awesome for any game that's even remotely westerny or apocalypticy.
2) Zoe Keating, Julia Kent, and Adam Hurst. All three of these folks play a mean cello. Not overbearing, but moody and tone-setting.
3) Vincent mentions Nick Cave and Warren Ellis in the book. I wholeheartedly agree. They have a compilation of a bunch of their stuff called "white lunar" that's a good Apocalypse World spin.
4) Max Richter. Contemporary classical artist who does nice, moody stuff.
5) Labradford. One of my favorite ambient acts. I find his albums Mi Media Naranja and Fixed::Context to be good Apocalypse World stuff.
6) Soundtracks by Clint Mansell tend to be good.
7) There's also a lot of stuff in the "doomjazz" genre that works pretty well. In particular, I like the band Bohren & Der Club of Gore. They have about a bazillion albums, each of which are just long chunks of real gloomy, melancholy, ambient jazz.
If I think of any more, I'll post them.