There were at least three main ways in which this happened.
First, there were many ordinary Americans who either personally were involved in the conflicts that arose from containment or who had family that was personally involved. We must remember that during these decades, the two major wars of containment, Korea and Vietnam, occurred. These wars touched the lives of many Americans who served in them or who had family who served in them.
Second, there was the nervousness and stress that the policy of containment brought about. Because the US was in conflict with the Soviet Union and was trying to contain it, there was always the fear of war. In the 1950s, for example, there were the “duck and cover” drills in schools and the ubiquity of fallout shelters. These sorts of things made life seem less secure.
Finally, it is possible to argue that containment helped to bring about the conformity that was seen in the ‘50s and the rebellion that arose in the ‘60s. Containment and the fear of communism encouraged people to conform for fear of being though un-American. The protests of the ‘60s arose in part in response to such attitudes.
Thus, the containment policy of the Cold War had serious impacts on the lives of ordinary Americans.