How did the threat of atomic warfare affect President Truman's Policy of Containment?
As the previous post outlines, in many ways it was in response to the fear of a nuclear conflict that would lead to incredible destruction on both sides. In some ways it led Truman and others to also be careful about where they were willing to get into conflict against the forces of Communism and where they were willing to actually commit US troops or where they wanted to fight strictly proxy wars.
Much of it also became about political influence within the United States. Truman could not be seen as soft on communism so he had to be able to articulate the way that he was going to "fight" them to appease the folks that were in favor of first-strike mentality and perhaps wanted to openly oppose Soviet moves. By articulating this policy he appeased them in some ways but also didn't lose the rather large group of people who wanted to be certain that we were not going to jump into a nuclear conflict that, in the end, would be lost by everyone involved.
Basically, it just made him have to be more careful.
If there were not a threat of having a nuclear war, it would be much easier to threaten the Soviet Union and to take chances in trying to drive the Soviets out of a particular place where they were trying to expand.
However, if there is the chance of nuclear war, you have to be a lot more careful. If all weapons are conventional, you can have a small battle here or there without too much risk. But if you have nukes, any battle can become a really horrifying war.