Seattle is a major city, and as when traveling to any other major city, one should take all of the safety precautions one would anywhere else. Keep note of your belongings, make sure your wallet or purse is not too easy to access, lock belongings in the trunk of your car.  Seattle does have its fair share of homeless and street people but in general they will not be a threat to your safety.  During the day, the downtown area is safe and popular areas such as Pioneer Square,  Denny Regrade, Pike Place Market, Seattle Center are usually populated with people. 

At night, use common sense.  Stay in lighted areas.  Try to be around other people.  Be aware of your surroundings.   Seattle Police are routinely on downtown bicycle patrols (day and night) so the tourist areas of downtown (3rd to I-5; Stewart to Cherry) are safe at night.  Pioneer Square and the Denny Regrade are known for their bars and clubs; visitors should exercise a little more caution late at night in those areas. 

The University District and Broadway are both magnets for homeless youth who can be intimidating but generally cause no harm.  Outside of these denser areas, Seattle is a fairly suburban, single-family home kind of city and most other neighborhoods are generally quite safe.