statue liberty
Radio Free Europe informs us the number of Russian applications for US asylum has surged again in 2016:
Russian citizens filed 1,912 new U.S. asylum application in the fiscal year ending September 30, up 31 percent compared to 2015 and 164 percent since 2012, when Putin was again elected president following a four-year stint as prime minister, new official statistics show.

RFE/RL obtained the U.S. Department of Homeland Security data after filing a request under the Freedom Of Information Act.
RFE also has an explanation albeit one it admits is pure conjecture based on "anecdotal evidence":
While the data does not indicate the basis for applicants' persecution claims, immigration attorneys link this multiyear surge to policies in Russia seen as discriminatory toward sexual minorities, a squeezing of dissent during Putin's third term, and widespread corruption.

Anecdotal evidence and internal accounting from lawyers working with sexual minorities suggest a sharp rise in the number of lesbian, gay, bisexual, and transgender (LGBT) individuals from Russia seeking U.S. asylum since Putin signed a 2013 law banning "propaganda of nontraditional sexual relations" to minors.
So Radio Free Europe believes Russian US asylum requests doubled in just a few years because gays in Russia are increasingly persecuted since the 2013 passing of what is usually dubbed simply "Russia's anti-gay law"

On the other hand actually only a handful of people have ever been fined under that law and almost always activists who had deliberately set out to violate it as a form of protest.

So I'd say there is another possibility. It goes like this: there are a fair number of people in Russia who would like to emigrate to the richer United States. In the past these would-be migrants would have known that being granted asylum in the US was highly unlikely and would not even attempted to emigrate using this path.

However, given the current tug-of-war between US and Russia, the increasing demonization of Russia in the west, and most of all the growing need of the west to believe itself morally superior to Russia, would-be migrants are guessing the chances of being granted an asylum are no longer as low.

What proof I have of this? None, like RFE's own explanation it's pure conjecture.

But here is one tiny hint: RFE does not say how many of these requests are approved and how many are turned down even by the US eager to believe massive horrors are being perpetrated in Russia. My guess: most of these applications are still turned down as economic migrants posing as persecuted minorities.