In the Korean War, China threatened to intervene if UN forces entered North Korea. UN forces crossed the 38th parallel (the pre-war border) 25th September 1950; within 3 weeks Chinese troops were moving into North Korea (Chinese advisors and observers were present months earlier than that; this was the first movement of large Chinese combat forces into Korea). Chinese troops started fighting in October, but their large-scale battles started in late November, rapidly pushing the UN forces south, and past the border.

What was the US counter to the Chinese entry into the war? After the initial fighting with the Chinese in late October and early November, it was clear that the Chinese had intervened in force - the question was in what strength. MacArthur's intelligence estimated a low 34,000 troops, and MacArthur felt he could continue to push north. The US Joint Chiefs of Staff didn't agree, and recommended that MacArthur immediately stop his offensive, and that a ceasefire should be negotiated as quickly as possible. Thus, the two conflicting paths of action were (a) MacArthur's continued offensive, largely ignoring the Chinese intervention, and (b) a quick diplomatic end to the fighting. The US National Security Council presented three options: (1) continue to advance, as MacArthur wanted, (2) withdraw back to the border, and be content with the successful defence of South Korea, or (3) stop and adopt a defensive stance. (2) and (3) were seen as harmful to US prestige, and MacArthur got his way, and continued to advance, resulting in the catastrophe of late November and precipitous retreat.

The UN command (i.e., MacArthur and his staff) considered the use of nuclear weapons against China, and also the use of radioactive material to cut Chinese supply lines. The Joint Chiefs of Staff prepared draft orders for nuclear attacks on Manchuria and the Shandong Peninsula to stop Chinese airstrikes from bases in those areas, if necessary. There were two problems with conducting major air attacks on China. First, the MiG-15 had been a rude shock to UN forces in Korea. Chinese air defences were credible, and US air attacks on China could result in heavy losses. Second, there was a high likelihood of Soviet intervention if the US attacked China - there was a mutual defence treaty in place which called for such Soviet intervention, and Soviet air forces were already provided defensive air cover over China.

Matthew Ridgway's superb leadership and prompt stopping of the Chinese advance, after replacing Walton Walker as commander of the US Eighth Army after Walker's death in late December, and then becoming commander of UN forces in Korea after MacArthur was relieved of that duty, took away the problem of stopping the Chinese - he'd done it. Neither the US, nor China, nor the Soviet Union wanted to expand the war to include fighting in China, whether in the air or on the ground.

In Vietnam, China supported the Viet Minh in their independence war against the French. Their support consisted of military supplies (weapons, ammunition, medical supplies, radios, etc.) and advisors and training. This aid began shortly before the Korean War, and the Korean War meant that there would be no large-scale Chinese military commitment.

As US involvement in Korea grew after the Gulf of Tonkin incident in 1964, Chinese military aid resumed. In response to US air strikes on North Vietnam, Chinese forces started moving into North Vietnam in July 1965. First, these forces included air defence units. Second, the Chinese forces were intended to deter a US invasion of North Vietnam. Chinese commitment reached a peak in 1967, with about 170,000 troops present. The Chinese intervention was limited, with no threat of invading the South, and while Chinese air defence troops helped, they were far from able to stop US air attacks. The US was unwilling to invade the North (there was significant domestic and foreign opposition to the use of US troops to support the South; an invasion of the North was not something that could be sold to the US public). Therefore, there was no need for the US to take any further action against China. Also, the Chinese presence was greatly reduced after 1968, due to (a) the Cultural Revolution, (b) worsening of Chinese-Soviet relations, and (c) Chinese desire to improve their relationship with the US.

For a lively discussion on China, the Soviet Union, and the US in the Vietnam War, see the discussion by [deleted], u/jvalordv and u/elblanco and u/BigBennP and others in https://www.reddit.com/r/AskHistorians/comments/236fa8/why_didnt_the_us_just_roll_over_north_vietnam_and/