Paranoia is fear based on unrealistic or illogical concerns. I'm not all that sure you can't get the USA to be much more paranoid about the Soviet Union and Communism than it was in the late 1940's and 1950s.
As others have said to create a cold war in which a "Red Dawn" fear was realistic, the changes have to occur outside of the USA. Ultimately, you have to create a USSR which was significantly more powerful, successful, expansionist, and truly revolutionary than the one that existed in the 1940's. You need a USSR that:
(1) Was far less dependent on US assistance in the Great Patriotic War
(2) Never really allied with the western powers and defeated Germany before the British and Americans invaded Europe.
(3) Became a model for much more successful series of local communist regimes in western Europe
(4) Was actually capable (in 1950-60) of fighting a stategic war against the USA and British Commonwealth (I'm presuming the UK stays in the US camp) - ie effective strategic bombers, a lot more missles, a lot more nuclear bombs, a real blue water navy, etc
(5) Was actually capable of projecting meaningful military power to assist revolutionary groups in the face of US power
(6) Did not join the UN (or whatever UN-like group exists), or dominates it
In our TL, the USSR never came close to the USA in military or diplomatic power until maybe the late 1970, and even then this was more because of US laxity than Soviet abilities. In actual fact, if a nuclear world war had broken out over Berlin, Cuba, whatever in the 1950's-1960's, the USA would have been able to destroy the Soviet Union with (as Dr Strangelove might say) acceptable losses. To have the US paranoia during that period be based on reality, you need a PoD in the early 20th century (or maybe even earlier) to create a Russia (and later USSR) that was industrially and technologically prepared to risk war with the USA much more than our USSR was. YOu probably also need to creat a Communist Party in the USSR that was far more revolutionary and far more willing to take risks than Stalin and his successors. An insane, brutal "big thinker" like Hitler in charge of the movement might help. Stalin met the first two criteria, but one could hardly give him credit for having the pure sweep of evil-thinking on a vast scale Hitler had.