I've been investigating ways to get from the airport to downtown Seattle. I'm staying at the Red Lion, and it looks like the best deal is the Gray Line Airport Express, which costs $10.25 (round trip $17) and makes stops not only at the Red Lion but also at the Crowne Plaza, Hilton, Grand Hyatt, and several other downtown hotels (for a list see here; the Red Lion is not listed, but I called there today and was assured by the bellman that it is served by the Gray Line). A taxi costs around $33 one way.
There is also a public bus (Route #194 Express or #174) which costs $1.25 (off-peak) or $1.75 (peak). The bellman at the Red Lion told me the nearest stop is 4 blocks from the hotel, so depending on how much luggage you have and how easy it is to move, that is the cheapest option.
Some of the larger hotels may have courtesy shuttles, which are free. Check your hotel's website to see if they offer this service.