Without a victory during the Korean War, President Harry Truman would probably not have won the United States Presidential Election of 1952 against Senator Robert A. Taft of Ohio. Heck, I don't think he would run for re-election at all. Truman was already in trouble from Joseph McCarthy's Red Scare and the exposure of corruption in his administration. If China joined the war and dragged it out longer, maybe General Dwight D. Eisenhower could run for President and win the primaries even against the establishment's Taft juggernaut. In OTL, Eisenhower was intimidated by the Truman's foreign policy victory in Korea, which was convincing enough for him to not run. Who the Democrats nominate is debatable. If Truman doesn't run, Vice President Alben W. Barkley would probably run, but he would be up against an open field of candidates.
I also remember reading about Truman's frustration with General Douglas MacArthur during the war, and how close he was to firing him. If the Chinese joined and prolonged the war, I think MacArthur would make enough provocative belligerent statements against China that would push Truman's buttons to their limits.