The United States did not become a member of the League of Nations because a group of senators opposed membership. They did not want the US to join the League largely because they did want the country to lose its ability to determine its own policies. These senators were unhappy with the idea that the US might be forced to enter wars to help other members of the League.
One of the aspects of the League of Nations was its commitment to the idea of collective security. This meant that the League was committed to helping its members if they were attacked. Many Americans were afraid that this would end up dragging the US into wars. Some Americans felt that the US should not have participated in WWI. They felt the war was not really important to the country’s interests. These Americans worried that another conflict in Europe (or elsewhere) could arise and that the US would be forced to go to war to protect other members of the League of Nations. Because of these worries, many senators opposed membership in the League and the US never ended up joining.