How did the Kennedy administration alter the direction of the Cold War, at least temporarily?
A part of the answer to this question could involve engaging in the mythology of President Kennedy. If we look at his Commencement Address at American University in 1963, one can see the groundwork and mentality that might have served to deescalate the Cold War. When Kennedy decries the notion of a "Pax Americana" and calls for cooperation with the Soviets, one can see powerful steps to articulate a common ground that would have minimized the Cold War Threat. Kennedy goes as far as to call on the Soviet Union and his own country to "reexamine the Cold War." In calling for the end to "the debate" that seeks to "pile up points," there is a genuine and sincere attempt to reduce the rhetoric that fueled so much of the conflict between both nations. The conclusion to the speech might go very far in pointing out that Kennedy sought to end the Cold War:
The United States, as the world knows, will never start a war. We do not want a war. We do not now expect a war. This generation of Americans has already had enough - - more than enough - - of war and hate and oppression. We shall be prepared if others wish it. We shall be alert to try to stop it. But we shall also do our part to build a world of peace where the weak are safe and the strong are just. We are not helpless before that task or hopeless of its success. Confident and unafraid, we labor on - - not toward a strategy of annihilation but toward a strategy of peace.
There is little to indicate that such words would have directly translated into practice and policy. There is even less to clearly state with authority that he would have ended the conflict. Yet, when examining the timing of these words in this speech, one is slightly persuaded to concede that had Kennedy lived, the ideas conveyed might have played some role in his practice.
Kennedy, having had sailors under his command die in the Pacific and having endured the death of his older brother over Germany during World War II, had an appreciation of what warfare and the rank of Commander in Chief meant. The major Cold War event during his tenure was the Cuban Missile Crisis in October 1962, where, as it is written, the Cold War almost turned Hot. Very hot, since both sides' missiles carried nuclear warheads, and full missile deployment would have guaranteed the destruction of all humankind. Although he took a "hardline" stance with this issue (presumably because Cuba was so close to the US), after the crisis was resolved he attempted to mitigate conflict with the Soviet Union and China, both of whom had embraced Communist ideology, and against which the US could not stand alone. Kennedy's "Bear any burden" speech reflects his approach to the Communist beliefs by fighting the opposition with words and ideas, and only as a last resort militarily, believing that the Pen was indeed mightier than the Sword. Had he remained alive and his attempts to remove advisers from Vietnam been realized, the US may have been able to avoid that war. After his death, the "hardliners" came to power, engaged in military conflicts against the Communists throughout the world, particularly in Vietnam, and brought about the awful, and well-known results.
I would say that the main thing the Kennedy administration did was to heat up the Cold War a bit.
Before Kennedy was inaugurated, there had not been any siginificant run-ins with the Soviets or a close ally of theirs since Korea. Eisenhower had been relatively conciliatory (the U-2 incident notwithstanding).
But then in the short time that Kennedy was in power, there were the two big incidents in Cuba (the Bay of Pigs Invasion and the Cuban Missile Crisis). You could also say that Kennedy's "bear in burden speech" and his talk about the missile gap helped to heat things up (as did his moves to increase US involvement in Vietnam).