I like to look at
http://spotcrime.com/wa/seattle for a general idea on current neighborhood safety. Looking at that website, while Belltown isn't the most dangerous area of the city, I wouldn't call it a safe neighborhood either.
In Belltown's small area (between Denny Way, Pine St. I-5 & the water) there have been 2 robberies and an assault in the past 2 months. Head a little NW to Queen Anne and there is nothing. I would personally rather live on Queen Anne than Belltown. However, Belltown is better than the Broadway area of Capitol Hill (between Roy St, Denny Wy, I-5 & 15th AVE NE) where there have been 3 shootings, 4 assaults, 2 robberies & 2 thefts in a smaller area.
Keep in mind however that this website doesn't take into account population density. Belltown is very dense with residents and employers, where as areas such as Queen Anne are not as dense.