It would be rather difficult for Ghenghis Khan to influence America very much, given that the United States didn't exist until about 500 years after his death. It would be over 200 years before Columbus made his first voyage in search of Asia. At the time the only contact between America and the rest of the world would have been the Norse expedition to Vinland, and that didn't exactly end well.

It's possible someone who's more knowledgeable about the subject could point to some cultural shifts that would affect America but with half a millennium of separation, Ghenghis Khan's influence on the USA would be pretty minimal.

There is one vicarious way I can think that they might be considered connected, and that is through the jingoist fear of what is called "Yellow Peril". YP is the idea that the Asiatic peoples are inherently expansionist and warlike, and therefore must be resisted with all appropriate military force. This concept became prolific during the lead up to WWII, and framed much of the US's relationship with the expanding Empire of Japan. It even manifested itself in the imprisonment of Asian american citizens in internment camps during the war. The origins of the Yellow Peril trace their spiritual roots to the western Mongolian invasions of Genghis Kahn in the mid 13th century. The Mongol empire and the subsequent khanates that followed plagued eastern Europe and the middle east for over one hundred years before finally fragmenting due to internal strife. This view of the invading mongols permeated itself into European identity, and was continued on in the consciousness new European nation on the American continent. I have only given a very rough outline to the topic, but if you are committed to examining the relationship between the Mongol Empires and America, this might be a way.