I don't think that Best Western is in THAT bad of a neighborhood. Just be careful at night, and be ready for lots of pandhandlers. It is in, or near, the Ride Free bus zone. The Silver Cloud suggestion is probably a little nicer neighborhood, though, and other hotels near there are probably fine, too- that's the South Lake Union area.
If you don't mind a ~1.5 mile bus ride north from South Lake Union, up Eastlake across the ship canal to Roosevelt Way, the Watertown and the University Inn on Roosevelt Way are both pretty nice. I think the Watertown is more expensive. There are regular bus runs along that corridor. These hotels are near the UW campus. They may be fuller during the week than on weekends, except during football season. There are some pandhandlers around there, too, though.
Seattle has a soft, crunchy liberal underbelly and is real "tolerant"
, and the climate is relatively mild, so there are lots of "people not living in conventional housing." You've been warned. I don't give them money, and don't think you should feel guilty about not doing so, either. There are plenty of resources for them to get assistance from.
Tourist season really cranks up after July 4 and is strong until mid-late August. Make your reservations early.
ETA: lotta good hotel leads posted above. I'll have to make note of them for any friends/family who come to town.
Quoted:
And while I've got your attention, what are some of the must see/do things in Seattle?
Ride a ferry across the Sound and back- Bainbridge Island or Bremerton; cheap and easy to do.
Lots of good restaurants- we can recommend some, and/or ask at your hotel.
Museum of Flight- can take a bus or cab to there. Has an SR-71.
Go to top of Space Needle- there is a restaurant /bar.
Top of Bank of America Tower- tallest building in town. That's a start...