The threat of atomic warfare affected the policy of containment by making it more dangerous and ensuring that it would need to be carried out in ways other than through direct military confrontation between the superpowers.
Before the threat of atomic war existed, it was much more possible to imagine wars between the Soviets and the United States. It was possible to imagine that such wars could be fought and won by one side or the other. This meant that it was possible to imagine containing communism through direct military conflict with the Soviet Union.
As nuclear weapons became more common and more powerful, the strategy of containment became more dangerous. The US had to try to prevent communism from spreading, but it had to do so in ways that would not provoke direct wars with the communists. The danger of this sort of thing can be seen in the conflict over the Soviet missiles in Cuba. The US wanted to prevent the spread of communist power, but their actions came close to leading to disaster. Thus, the threat of atomic warfare made containment more dangerous and made it imperative that alternatives to warfare be found.