So.... a lot has changed in Seattle in the last few years. The almost complete renovation in Belltown is one of them. When I first moved here there were still some pretty seedy spots in Belltown. Now is almost all high-end condos. But here's what it used to be like:
For the most part, Seattle is a safe city. Just like all cities, there are places you avoid late at night. The Belltown neighborhood has a lots of restaurants and bars, and, as a tourist, you may be patronizing one of them. If you should find yourself at a bar until closing time, needing to walk back to your hotel, stay off of 3rd Avenue. 1st is great, and 2nd and 4th are OK, but parts of 3rd can be a bad news. It's even a little scary during the day, but can be dangerous late at night, when no one is around.
Recently (October 2000) there have been several gang muggings in Belltown, a hip Seattle neighborhood that is noted for quality restaurants.
Personally, I always travel with at least a folding knife carried at ready. Not to be paranoid, but, as in any city, caution is the best approach and being armed is wise.