Eisenhower, it should be noted, refused to give air support to the French during their defeat at Dien Bien Phu in 1954. He felt it was a colonial war, not a war of containment, and knew the public wouldn't support it so soon after the Korean War. He did send aid and advisers to South Vietnam, but the insurgency did not start for real until after 1960 and Kennedy's election.
Kennedy carried on with Eisenhower's policies, in part due to his distraction with foreign policy events in Europe and the Western Hemisphere, with the Berlin Wall, Bay of Pigs and Cuban Missile Crisis. Vietnam was off most peoples' radar screens, including his.
LBJ made the determination to fight the war, asked Congress for a free hand and received it (Tonkin Gulf Resolution) and began a four year escalation of troops and offensives that would culminate with the Tet Offensive and LBJ's decision not to run again in 1968.