They basically have to deploy in Korea. The Japanese want the US forces out, because of all the rapes in Okinawa, and the fact that the US won't let the suspects stand trial in Japan.
If they deploy in Taiwan, the PRC will go absoultely nuts (think along the lines of USSR putting missiles in Cuba).
After that they're pretty much out of friends in Asia.
They won't deploy in Mongolia because:
1. The mongolians won't let them. The only reason the US has bases in Asia at all is because of WWII. No base then = no base now.
2. it's only just stopped being Communist, and may swing back that way (too scarey for the US).
3. It would piss off China, North Korea, And a whole host of former USSR states. All armed with nuclear weapons. And a base in the middle desert in a country half the size of Western Europe and a population of 1 million - definite tac nuke risk...
So Korea's about the only option they got.