That would be JFK and increased by 500,000 men after the 1964 elections by one LBJ. 

Of course Eisenhower came up with the "Domino Theory" of Anti-communism, but JFK did the Bay of Pigs and did the big Vietnam start.

Truman sent us to Korea and Eisenhower did the cease fire there.

 Nixon was the one who ended Vietnam, eventually.