The fear of a communist takeover was not so much of a military invasion as it was that the American way of life would be supplanted by an insidious communist plot. At a time when conformity was the norm, those who spoke against the government were accused of being communist sympathizers. J. Edgar Hoover questioned the patriotism of such individuals in his book None Dare Call it Treason. In response to the perceived threat, "In God We Trust" was added to U.S. currency and coinage, and "under God" was added to the pledge of allegiance. It would be unfair to suggest that Americans suffered from paranoia. The communist doctrine was for the revolution to encompass the entire world. Many in America saw telltale signs of that revolution beginning in this country and were determined to stop it.
First of all, please note that the fear of communist takeover of the United States is usually much more connected with the McCarthy Era of the 1950s, not with the 1960s. Secondly, the answer to this can only be a matter of opinion.
My opinion is that an actual communist "takeover" of the country was impossible. Americans' attitudes about government and economics are much too firmly anti-communist for such a takeover to have happened. Americans would never have allowed a politician or small group of politicians to implement communism. Even today, long after the Cold War, we can see how strongly people have reacted to "Obamacare," which is hardly a communist takeover.
However, it is important to note that there were communists in the United States who were pro-Soviet. The Rosenbergs really were spies. Alger Hiss was likely a Soviet agent. There were truly communists in America, but there was no way they could ever (in my view) have taken over the US.