Well, I'm drawing from Khrushchev's memoirs here, so not the most unbiased source.

However, I think it's one worth hearing, as we in the west rarely get the Soviet perspective on events of the time.

In the mid-late 1950s, the US started basing medium range nuclear missiles in Turkey. This was worrying to the USSR not only because those missiles could conceivably strike targets in the USSR before the Soviets could muster a response, but because it also represented further tactical encirclement of the Soviet Union.

Khrushchev had wanted the missiles out of Turkey for a long time. Even back in the Eisenhower days, he was constantly trying to get them removed.

The problem is, he really had no means of putting leverage on the US. That was, until the Bay of Pigs essentially forced Castro into the Soviet sphere. At first, Castro was rather ambivalent about the Soviet Union. So much so that he actually flirted with the Non-Aligned Movement.

However, once it became clear the US would not leave him alone, he had no choice but to cozy up with the USSR, as they were the only power which could credibly protect him from the US.

So, once Castro drifted into the Soviet camp, Khrushchev saw a chance to twist the US' proverbial arm about the Turkish missiles. He honestly did not expect Kennedy would react the way he did to the placement of missiles in Cuba.

His whole plan was to use the missiles in Cuba as a way to get the US to the table, and while it eventually worked, it caused a lot more drama than Khrushchev intended. Tit for tat had been the rule of the Cold War, and so Khrushchev thought Kennedy would accept the placement of missiles in Cuba as such, making him more amenable to negotiation.

I get the impression from reading his memoirs that Khrushchev viewed Kennedy as an arrogant, aristocratic brat who had never had to take no for an answer his whole life. Nikita was appalled by Kennedy's stubbornness and brinkmanship, and viewed him as a dangerous man who was more concerned with winning and appearances than the well being of his people.

In the end, Kennedy finally negotiated, but due to the missiles being dismantled and removed from Turkey in secret, whereas the Soviet missiles were publicly removed from Cuba, it blew up in Khrushchev's face and was one of the factors that ended up costing him his job. Kennedy kept his word, but in the eyes of the public, he "won" the Cuban missile crisis.

This meant the thin ice that Khrushchev was already on with the Stalinists in the politburo began to crack. It's arguable that had Khrushchev handled the aftermath better and retained power, the USSR would have been a much different place.

Now, who was more responsible? I'd say Kennedy myself, but that may be because I generally agree with Khrushchev's views on him.

The answers here already are valid, but there's information to add on the other hand too:

Khrushchev's foreign policy was built largely on the principle of using low level aggressiveness to hide Soviet weakness. The metaphor he used was the meniscus on a glass of wine - the USSR should be pushing the West, but not enough to spill over into war.

Kennedy and Khrushchev first met in a conference in Vienna in June 1961. Khrushchev publicly embarrassed Kennedy. He was much more aggressive and accusatory in front of the cameras and made a much better public presentation. Khrushchev then had little respect for Kennedy and though of Kennedy as a pretty boy pussy who he could push around. This, in part, led him to overplay his hand. He thought Kennedy would do nothing. In fact, Kennedy knew that the US had a tremendous nuclear missile advantage over the Soviets. In short, Khrushchev was trying to bluff someone who knew what his cards were.

Source: Aleksander Fursenko, Khrushchev's Cold War: The Inside Story of An American Adversary (2006)

What exactly do you mean by "more responsible"?