There were multiple reasons as to why the US didn't want to join the League of Nations.
1) Many Americans and members of Congress wanted to stay out of European affairs. They would much rather go back to a policy of isolationism, in which the US does not interact with European nations.
2) The power of Congress would be decreased. If the US joined the League of Nations, then the League of Nations would be able to decide when a country goes to war. Congress didn't like this idea at all, as they are supposed to be the legislative body in America that decides whether or not the US goes to war with another country.