There is no way that Obama hasn't taken the temperature on SOPA by now. Obama's most vocal liberal base will transform into an internet hate machine to rival the likes of the tea party if SOPA is signed. There is also the Ron Paul crew capturing more and more Democratic voters by the day, a signature on SOPA would be an incredible windfall for them in terms of people who would refuse to vote for Obama out of pure anger.
I don't underestimate the influence of these interests in Washington, but SOPA is really unpopular right now and getting more so by the day.