The fears of the Cold War were closely intertwined with the fears of atomic and nuclear power, and the implications of the arms race between the US and Soviet Union.
At the end of World War II, it should be remembered that, for a very brief time, the United States held a monopoly on atomic power, but in 1949, the Soviets successfully tested their own atomic bomb. From this point forward, both polities had access to weaponry of mass destruction and, furthermore, in the years that would follow, they would both continue to advance their weapons technology to create more powerful and devastating weapons in later years (such as the hydrogen bomb, for example, or the ICBM). What resulted was a nuclear arms race between the two, as each tried to outpace the other, unable to accept a military disadvantage against its rival.
This contributed to the fear and paranoia present within the United States where the Soviets were concerned. For one thing, there were very real legitimate concerns about espionage and Soviet attempts to steal state secrets, but additionally, the potential for mutual destruction cultivated an underlying atmosphere of sustained terror within the population as well.
It would have been practically impossible to have lived in the United States during this time period without being aware of the possibility of complete atomic destruction (and indeed, the theme of atomic destruction serves as a critical component of the science fiction and pop culture literature of the Cold War Period. All one needs do is read the works of a writer such as Ray Bradbury or watch television programs such as the original Twilight Zone to see the degree to which nuclear annihilation was front and center and consider what it would have been like to have lived in a time period where this kind of mentality loomed so large). The communist Soviet Union was, then, not only an ideological opponent, but, within this nuclear context, it would have been understood as a potential existential threat, which only would have heightened and intensified the fear and paranoia of the era.