The United States should boycott the Olympics. Not just the forthcoming Winter Olympics in Sochi, Russia and not even the planned 2016 Olympics in Rio de Janeiro, Brazil. It should boycott the Olympics themselves. Whatever noble purposes the "Olympic movement" might have served in the past, they have passed. The Olympics have no economic benefit, no true benefit to sport (at least relative to alternatives), prove nothing for the nation, and serve to legitimize authoritarian regimes. The United States government shouldn't support them.
Economics first. While cities and countries sometimes host the Olympics with the hope of major economic benefits or "nation branding" these things almost never materialize. With the exception of Los Angeles in 1984--a special case because the city was the only one in competition and managed to get the games without building many new facilities--the Olympics has never turned a profit. Otherwise, Olympics have left a trail of red ink. Although the U.S. spends almost no federal dollars on its Olympic efforts when it doesn't, host many other countries do and in so doing, they subsidize something (sports) that seems to work just fine as a market-drive activity.
The Olympics don't advance the cause of sports themselves either. Most Olympic sports are already quite popular as either spectator (basketball, ice skating) or participant (mountain biking, running) activities. While it's possible that a few activities like discus and luge could fall by the wayside without Olympics, I'm not sure who would miss them. Even if one wanted to spend it on sports, the vast sums of money to promote and train Olympic athletes in these obscure disciplines, furthermore, could almost certainly better be spent on efforts to expose more people to them on a regular basis.
The United States also has nothing left to prove. Since a Soviet team competed for the last time in 1992, the United States has won the most total medals in every single Summer Games that has taken place. We have also placed either first or second in total medals at all but one winter games since then. There's little incremental national pride to be gained.
Finally, the Olympics help to legitimize some of the world's worst regimes. All three of the greatest mass murdering regimes of the 20th century--Nazi Germany, Soviet Russia, and Communist China--have hosted the Olympics. Fascist Italy and Militarist Japan, the other two major World War II aggressors, also both put together successful Olympics host proposals (although the actual games took place in both countries after they had established democracies.) Given that most truly repressed countries simply won't have the money or willingness to allow scrutiny that come with an Olympics, it seems like dictatorships and authoritarian states have a leg up on hosting the Olympics. Worst of all, the Olympics gives an equal station to regimes that simply don't deserve it. The cruel family-based totalitarian dictatorships of North Korea, Syria and Cuba all get equal status with-established democracies on the Olympic venues. When they run decent training programs, like those of Cuba and the former East Germany, furthermore, totalitarian regimes can perform far better in the Olympics than raw population numbers would suggest. This can build domestic support for dictators. The 2008 Olympics did just that for China's rulers and the 2014 games will probably do that in Russia.
Given that the Olympics is largely a private undertaking in the United States, there's not much the government can or should do to stay out of the Olympics. Banning athletes from traveling to the Olympics (as Jimmy Carter did in 1980) is unfair. But not a single federal penny should flow to efforts to host the games in the United States and high government officials should simply ignore the games in Russia and discourage athletes from going. In the end, there's no reason the United States should participate in the Olympics.