As far as I know, there are no nonstop flights from the US to Bangkok, which is perhaps expected given the distance to Thailand from the US. Thai Airways used to fly nonstop from LAX to BKK, but that flight now has a stopover in Seoul.
Luckily, there are numerous options to get to Bangkok on Star Alliance carriers. Bangkok is the hub for Thai Airways, and they’re renowned for their ground service at BKK. Other Star Alliance carriers that have flights to Bangkok include Air China, ANA, Asiana, Austrian, Egypt Air, Ethiopian, EVA Air, Lufthansa, Singapore, Swiss, Turkish, and United (i.e. nearly all of the carriers in Star Alliance that you’ve heard of).
The obvious choice if you’re flying out of BKK, particularly in a premium cabin, is to fly on Thai Airways. If you fly First Class out of Bangkok, then you get access to the Royal First Lounge and Spa, where you get a 1-hour massage (business class passengers get 30 minutes). Thai currently has First Class service to Sydney, London, Frankfurt, Munich, Paris, Madrid, Rome, Zurich, and on some flights to Hong Kong and Tokyo.
Thai is also great because they regularly release a lot of award space, both far out, in the interim, and close in. For example, there’s premium cabin award space on nearly all dates for flights from London to BKK for the entire schedule that’s loaded. Same with Frankfurt and Hong Kong. Tokyo is harder, but still easy if you’re planning at least a couple of months out (although you’ll be harder pressed to get First Class space on the A380 out of Tokyo).
Part of the reason why Thai releases so much award space is that their in-flight product is generally considered to be a notch below that of airlines like ANA, Asiana, Lufthansa, Singapore, and Swiss. The hard product is good on the A380 and the new suites on the 747 (and it’s fine on most 747s, although there’s one plane with an older version that you want to avoid), but people criticize the inconsistent soft product. But again, if you’re looking for flights departing out of Bangkok, I think you should seriously consider flying Thai for the first leg so that you get the full ground experience at BKK.
Since there are no nonstop flights to BKK from the US, you have to connect, and given how many Star Alliance carriers have flights to Bangkok, you have tons of options available. From the east coast of the US to Bangkok, it’s roughly the same distance to route through Europe as it is through Asia, and both United and US Airways allow routings through Europe to get to Southeast Asia. The most “aspirational” routing would probably go through Frankfurt on Lufthansa, but Lufthansa doesn’t offer First Class award space more than 15 days before departure. Routing through Asia, the most aspirational routing in First Class would be ANA through Tokyo, which is also a hard award seat to snag. A good alternative is to book legs on United to make sure that you have a seat assured and then switch to Lufthansa or ANA if the award seat opens up closer to departure. If you don’t have status, you’ll have to pay change fees, but that can be worth it to experience a significantly better product for a 10-hour flight.
United charges 60,000 miles for a one-way to South Asia in Business Class, while it’s 70,000 miles for First Class. US Airways charges 120,000 miles for a roundtrip to South Asia in Business Class and 160,000 miles in First Class, although you can likely get lower amounts charged if you say that your trip to Bangkok is a “stopover” on your way to a destination in North Asia, or you can continue on to Australia or New Zealand and pay fewer miles.