When I studied this part of history in high school the attention was put more on the Hollywood actors, playwrights or scientists who were accused of Communist, but rarely Russians living in the United States at the time. Were they primarily anti-communist at the time, which saved them from the attention of McCarthyism et al, or was it something else?
This is absolutely not to detract from any other possible answers, but some useful background might be found in this answer I wrote about Senator McCarthy and the Red Scare (which properly speaking went from about 1945 to 1954).
At least in the context I looked at, the two major areas of fear among McCarthy in particular were Soviet spies, and Communist Party members and their sympathizers. What's interesting is that the targets of these hunts if anything tended to be elite, Protestant, Eastern Establishment figures (what we would call WASPs), or particular groups that were seen as decadent or open to blackmail, namely Communist sympathizers in Hollywood and homosexuals.
So while I wouldn't say that ethnic Russians or other eastern Europeans were never targeted - the United States at the time had massive pressure for those from such communities to assimilate into the cultural mainstream - they do not seem to have been the prime public targets of the scare. You were much more likely to be a suspect if you were, say, a Hollywood screenwriter who had participated in Communist-affiliated popular front politics in the 1930s, or were a government employee suspected of homosexual leanings, or were an upper class Harvard grad like Alger Hiss.