I live in Queen Anne which is about a mile up the hill from downtown. It’s convenient for hopping on the bus and heading downtown. Another area close to downtown that you might want to take a look at is Capitol Hill. As for downtown areas, I’ve heard some areas are safer than others. You might want to avoid First Hill, Pioneer Square, and parts of Belltown. Have you taken a look at the crime stats? I moved from LA so was pleasantly surprised at how much less crime there is here in Seattle by comparison. Where are you moving from?

http://www.cityofseattle.net/police/crime/default.htm