Answer: Obligate the US to fight in future wars.
Details:
The majority leader of Senate, Henry Cabot Lodge, argued that the commitment to the League of Nations (a part of the Treaty of Versailles) could take away US Congress' constitutional right to declare war and obligate them to military actions initiated by the League of Nations.
The Republicans in the Senate feared the treaty could commit the US to future wars that were not directly related to US national security, because of the commitment the treaty made to the formation of a League of Nations.
The United States never joined the League of Nations, in spite of the fact that an organization such as the League of Nations was the signature idea of US President Woodrow Wilson. He had laid out 14 Points for establishing and maintaining world peace following the Great War (World War I). Point #14 was the establishment of an international peacekeeping association.
The Treaty of Versailles adopted that idea, but back home in the United States, there was not support for involving America in any association that could diminish US sovereignty over its own affairs or involve the US again in wars beyond those pertinent to the United States' own national security. Because of its objections to membership in the League of Nations, the United States Senate refused to ratify the Treaty of Versailles.
