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I think the idea that boycotting the Sochi games would have no effect is ridiculously naive. Would boycotting Sochi change the Russian anti-gay policies right now? Propably not. The long-term effects are however a very complicated matter.

The world is not a place where actions are isolated; everything affects everything.

LGBT rights have become one of the main human rights battles of the first part of this century. The LGBT issue, like most human rights issues, are ultimately a question of public opinion. Public opinion is historically often moved by example, and respected organizations and public figures taking a strong stand on issues can have a huge impact. Many people will notice if there are boycotts over these issues in the olympics, and many of those people will be affected by it to some extent.

This goes both ways too: respected individuals such as athletes showing up as if nothing is going on and is also sending a message; that LGBT rights are not worth making a fuss over.

Also, public opinions are swayed by politicians, who are affected by money. Boycotts are bad for the sponsors. They take eyes of the product and they put stains on the image they're trying to build.

You can't win a human rights battle overnight, but doing the right things many times over will eventually move things the right way and make persecuting LGBT minorities a less worthwhile pursuit for power-hungry politicians looking for an easy target.