John Kennedy quantifiably escalated the war in Vietnam over the course of his presidency. He inherited American involvement in the conflict from Eisenhower, who had guaranteed South Vietnam's continued existence after the partition of Vietnam at the Geneva Accords in 1954.

In brief, the French tried to hold on to "Indochina" (Laos, Cambodia, and Vietnam) after the Second World War, but local resistance drove France out of the region. After the French withdrawal, world powers gathered in Geneva in 1954 to discuss how to stabilize the region. The conference, which also tried but failed to bring an end to the Korean conflict, determined that Vietnam would be divided into North and South until elections could be held. The United States, however, never signed the agreement and instead worked to prevent a complete takeover of Vietnam by the Soviet Union's Communist allies in Hanoi. After an insurrection emerged in South Vietnam between 1954 and 1959, Eisenhower committed around 900 military advisers to help South Vietnam put down the Communist-aligned insurrection.

Thus Kennedy, like his predecessor Lyndon Johnson (who would escalate American involvement into a full-blown American war in 1965), inherited a choice regarding Vietnam--to withdraw, to continue to maintain the current advisory role in Vietnam, or to commit American troops to fighting in South Vietnam. Kennedy made clear his point of view on the matter: the United States would not abandon South Vietnam--and by extension, the region--to a Communist takeover, but it would also not commit to fighting the war for the South either. But while JFK never called for direct military intervention in Vietnam, he increased the American troop presence in Vietnam from 900 to 16,000 over the course of his presidency. Scholars have argued ever since whether JFK would have eventually pulled out of if he would have escalated the war like LBJ did. Some of the arguments are rather obnoxious, I would say. There is one book published by a university press (I can't remember the name right away, but I could look it up for you) that goes into detail in its introduction to convince its readers that it is not doing counter-factual history but something far greater--reconstructing what JFK's actual response to certain events would have been over the next few years based on an analysis of JFK before his death. Nevertheless, any analysis of what JFK would have done in 1964 (an election year) or 1965 (after theoretically having been reelected) is complete counterfactual history, akin to predicting the future in other scholarly fields. The short of it is this: JFK objectively and numerically increased American participation in the Vietnam War during his presidency, though you could take either side in the argument over whether he would have escalated American involvement further.