Here is what Nina Tannenwald reports on the polling:

Public opinion polls supported the link between public opinion and an emerging taboo, showing the public supporting some, but not all, uses of atomic weapons. Polls taken during the Korean War show that American public support for using atomic weapons against North Korean forces never went above 20 percent, even though North Korea did not possess nuclear weapons. However, the American public did support using nuclear weapons against China after it entered the war. Initially, in 1950, only a minority (27 percent) supported using atomic weapons against (non-nuclear) China. However, after China entered the war, the public increasingly supported their use against China in various circumstances. Toward the end of 1951 — a year after Chinese forces had intervened — a Gallup poll reported that 39 percent of respondents supported use of the atomic bomb “on enemy military targets in Korea” by United Nations forces, while 10 percent gave it qualified approval. In 1953, 45 percent supported using atomic weapons against Chinese cities if the United States got into a war with China, and a majority (56 percent) supported using atomic artillery shells against China if peace talks broke down. Thus the public supported using nuclear weapons first, but only against the Soviet Union or China, and not against North Korean forces even though North Korea was non-nuclear. As the State Department’s polling showed, world and allied opinion was even more strongly opposed to using atomic weapons than was the American public, largely because European allies felt they would be the battleground in any atomic war between the superpowers.

Nina Tannenwald, The Nuclear Taboo: The United States and the Non-Use of Nuclear Weapons Since 1945 (Cambridge University Press, 2007), 151-152.