The Vietnam war, as we know it, for all intents and purposes, began under Kennedy, in 1961.
That's debateble. Some say it was Ike Eisenhower who started it. This is an academical question for the Vietnamese, since USA just continued France's war.
Not that I'm defending JFK or anything.
Eisenhower sent in the first troops, though granted he didn't intend for it to be a war. And JFK started to withdraw the troops before his death. If anything, it was Johnson's war.