I would recommend checking out the many options that exist in the way of the rent-controlled housing for low-income residents. With your income you may qualify.
A common rule of thumb is to spend a quarter of your income on housing, so living on the cheap side puts you at about $600-$700 per month.
Market conditions seem good right now for renters, with relatively stagnant or lowered prices (I just moved out of a 1 bedroom on Capitol Hill after living there almost 2 years, and they raised the advertised rent $25) and lots of places offering one month free rent and other perks.
As far as overall safety, almost all of Seattle is safe. Belltown gets a bad rap, but it strikes me as more of a sketch/crazy/transient factor than genuine lack of safety. The Central District also gets a bad rap for gang activity.
Belltown: expensive for the masses, see above re: safety. Lots of low-income housing here if you qualify, though.
Capitol Hill, Queen Anne, or Fremont: max out your city living cred. You could affordably rent a room in these neighborhoods, or studios can also be found here for $700. Amazingly, I'm seeing some one bedrooms below $700 on Capitol Hill right now. For a better-quality place (larger, more recent construction), think north Seattle.
Wedgwood, Ravenna, Phinney, Wallingford, Greenwood, Green Lake: this seems like your sweet spot, unless you are really set on a walkable city life in which case I strongly recommend you go for a cheap Cap Hill studio/1br.
West Seattle: similar to north Seattle neighborhoods above, but more isolated from the rest of the city. Safety level and rent are going to drop as you move south towards White Center / Burien.
Lake City, Northgate, Central District, Rainier Valley: very affordable, but among some of the less safe neighborhoods.