The Cold War helped to make the United States less democratic by making it less protective of people's rights. This can be seen most clearly in the era of HUAC and McCarthyism. There was little or no tolerance for the expression of viewpoints that might be seen as sympathetic to communism.
The Cold War helped to make the US more democratic by helping to promote acceptance of civil rights for African Americans. Many American leaders felt that the US had to improve its race relations so as to look good to the non-whites in the Third World. Because of this, there was more support for civil rights than there might have been if the Cold War had not been going on.
Seems to me that democracy has nothing to do with rights, tolerance, or expressing viewpoints. Democracies make decisions by majority vote without regard to rights, tolerance, or expressing viewpoints.
With that in mind, if Joe McCarthy’s (and Lyndon Johnson’s) view of communism represented the view of the majority, then that would be democratic decision making. However, I have no idea if the majority agreed with McCarty and Johnson.