Asylum is for people who are afraid to return to their own country because they fear persecution, unlawful imprisonment or even death because of their race, their ethnicity, their religion, their membership in particular social or political groups, or their political beliefs.
Mr. Snowden undoubtedly fears returning home because he would be arrested and prosecuted. But those fears do not qualify him for asylum. And does he really feel safer in a country where Mr. Putin, an increasingly authoritarian leader, has jailed and persecuted his critics?
via NYTimes

This is complete newspeak on part of NYT. Mark the last words in the first quote “their political beliefs”. The case of Snowden is not about military secrets, but about his political beliefs. The belief that those in power should not abuse it, the belief that those who have abused the power should be brought to light. It is in fact for these very beliefs they are targeting him.
And why should not he fear arrest and prosecution? As they have done with Manning, they will do with him and Assange. This would be just to set an example, so that no one else does it. Actually Putin and Obama are no different. If at all someone from any other country, lets say Cuba, would come to the US, having leaked Cuban secrets, won’t the US consider giving them asylum. And does giving that person the asylum, has to do anything with how Obama himself is running the show. If spying on your own people, breaching their privacy to the fullest is okay then jailing and persecuting the critics is no different.