The unit that dropped the Atomic Bombs on Japan was the 509th Composite Group. It had no connection to Minot Army Air Force base. When it was formed in December 1944, Col. Paul Tibbets chose Wendover Army Air Field, in western Utah, close to the border of Nevada, as the location for the 509th Composite Group's training.
Source: "Return of the Enola Gay" by Paul Tibbets (1998)