The initial response was similar to Assad's to protests in Syria: the protesters might have legitimate grievances, but are cautioned not to cross over the line. That quickly proves inadequate; the next step is to blame the West. That is happening in Russia now.
To those who care about Politics in Russia, I can tell that the vast majority of Russians, at least those outside of the major cities are still scarcely aware of the protests which got scant attention on the national news. The biggest news item was a mistake by Fox news, which in place of a peaceful protein in Moscow showed a violent scuffle in Athens.
Masses never had much to do with guiding the revolution, so this ignorance may not prove decisive; however, many opinions can still be shaped by government propaganda. Those who are aware are annoyed by being belittled in this way. There's no way to keep a lid on the truth, and that leaves the only recourse to the unscrupulous - Force. Putin is still hoping that Lies might suffice to clam up the popular discontent. The Russians tolerated a lot, but the cynical exchange of jobs between Putin and Medvedev is too much for many to accept. People know they are facing 12 additional years of Putin's rule. The 'blame the West' strategy worked poorly during the Soviet times, now it is widely seen as a crude ploy by Putin and the United Russia to avoid scrutiny.
It's worth describing the tortured logic. "There is a well-organized pattern of destabilizing society," Putin said in a call-in TV show, referring in part to Hillary Clinton's call for new and fair elections.
What pattern are they talking about?
In their alternative universe Quaddafi was a victim of Western aggression, a legitimate leader whose 'assassination' Russians protested in the UN. When Quaddafi was on the ropes and out of friends, he could count on political cover from Russia. Another example is Syria.
Why would the West do this? "They still fear our nuclear potential," Putin said. "We also carry an independent foreign policy, and, of course, it's an impediment for some." An impediment! Russian policy since before WWII has been nothing but a continual series of impediments to peace and justice.
Fortunately, I can say with certainty (at least for large metropolises) that Putin's words are counter-productive, because they do not pass the laugh test.