The area around the airport is not appealing for a tourist, unless you have no other choice (really early flight on the last night of your stay, for example). Staying anywhere downtown, or in some close-in Seattle neighborhood like Queen Anne or Capitol Hill, would be tons better.

What would be the closest place to stay near downtown? Is Queen Anne or Capital hill walkable to downtown?

Lower Queen Anne and the west side of Capitol Hill are within easy walking distance of downtown. I've lived in both neighborhoods, and I prefer the walk from Capitol Hill. For instance, if you stayed at the 11th Avenue Inn Ben and Breakfast on Capitol Hill, you would have three interesting walking routes to downtown: Pine, Pike, or Olive Way.

HTtY

PS If you are a senior, you can apply for a card that will give you $.75 fare on busses which come with a transfer that's good for at least 90 minutes.