There are several neighborhoods I would suggest but those are really dependent on your interests and every Seattle resident is going to tell you that their neighborhood is the best. They are also all about 5 minutes of driving from one neighborhood to the next so that is not exactly convenient for someone that only has one day. The best way to make the most of your day is to go downtown. From there you can walk to Pioneer Square (Seattle Underground, Elliott Bay Books, historic, touristy, expensive galleries), Belltown (boutique shopping, fine dining, bars, interesting galleries), the waterfront (Olympic Sculpture Park [stunning], good views, touristy, a few good places to eat), Queen Anne (Experience Music Project, SciFi Museum, Space Needle, the Mecca, Pacific Science Center), or stay downtown (Seattle Art Museum, Barneys, Marios, Butch Blum, Nordstrom Flagship, Rem Koolhaas library [must see], incredible restaurants, go to the top of the tallest building in town, get a latte at the worlds highest Starbucks, get a latte at the worlds oldest Starbucks (Pike Place Market], lots of other stuff...). You can walk to all of these places from the center of town. There are buses that run from Sea-Tac to downtown several times an hour. If you want something else you need to go into the neighborhoods, which is really where Seattle is at. If you can't find anything of interest in the list above let me know and I'll give you a rundown of the neighborhoods if you can be any more specific about what you want to do.
edit: None of the options in bold type will be more than about 15 blocks from where the bus would drop you off, so, walking distance. You could take a taxi but that would be a waste of money. As others have said, it would be cheaper to rent a car and pay for parking. A car would also afford you the opportunity to see some of the things that make us all love Seattle, most of which are not in the vicinity of downtown.