What were the most important parts of US policies of containment in/to Asia in Korea and Vietnam?
The most important part of "Containment" was not the most obvious. Economic aid to anti-communist governments was important in building up the economies and allowing the positive propaganda of improved standards of living, consumer goods, etc. This also built up approval of the local government by the populace, and built up the stability of the government. This succeeded long term in South Korea.
Military aid, even to the extent of using American troops in Korea and Viet Nam, as well as troops as advisors and trainers in many countries was also part of the strategy. These countries had the benefits (sometimes dubious) of our military advice and assistence, and access to weapons in quality and quantity they could otherwise have never afforded.
The most important aspect, though, was the idea that the Soviet Union could not sustain the World Revolution indefinitely, and that eventually their economy would crack. This was the idea of historian George Keenan. In his obituary, the New York Times wrote:
As the State Department's first policy planning chief in the late 1940's, serving Secretary of State George C. Marshall, Mr. Kennan was an intellectual architect of the Marshall Plan, which sent billions of dollars of American aid to nations devastated by World War II. At the same time, he conceived a secret 'political warfare' unit that aimed to roll back Communism, not merely contain it. His brainchild became the covert-operations directorate of the Central Intelligence Agency.
Keenan was later disappointed when much of his idea was subverted by the massive arms buildups of the 1950s and '60s, but he was correct. Former President Richard Nixon pointed out early in the Soviet-Afghani war that the USSR's economy was damaged beyond repair, and their inevitable military failure would destroy their already teetering economy. This proved to be the case.
Many people ascribe the "victory" of the US in the Cold War to Reagan, but he was the recipient of the benefits of Carter's decisions to rebuild the US Special Operations capabilities and use them and the CIA and Defense Intelligence Agency to aid the Mujahadden. Containment failed to stop the spread of communism, but it did succeed eventually in destroying the Soviet Union. But the process far predated Reagan or Carter, being instituted originally by President Harry Truman. Reagan's success was the result of 45 years of overt and covert containment.