What are some reasons why the United States should not have entered the Korean War?
The United States entered the Korean War because it felt that the war was a threat to its security. It felt that a communist victory in Korea would lead to further expansion of communism. It felt that a further expansion of communism would be a danger to the United States. If you want to show that US entry into this war was not necessary, you need to argue against these points of view.
You could argue that communism as a whole was not much of a threat. You can say that there was no way that the communists were going to seriously threaten the US. Therefore, there was no reason to spend resources and lives containing them.
You could argue that communism was a danger, but that the Korean War was not important. From this point of view, you can argue that a communist takeover of Korea would not have led to any further spread. Just because Korea became communist does not meant that Japan or the Philippines would.
By making these arguments, you would be able to call the American rationale for entering the war into doubt.