In college sports, the idea for an athletic director is to have strong programs in all sports that your university competes in. They key is to hire great coaches to build and oversee those programs and orchestrate their success. Yet the two big sports that have always been vital are basketball and of course football. Chances are if your athletic program has both a good gridiron and hoops program, you're going to find yourself in stable security as an athletic director.
With this list, the main factor in listing the top football/basketball schools was balance. Both programs have to have successful and have widespread appeal. There are some schools where one sport is just a tad more alluring, but not like at Duke, where the hoops program is the end-all-be-all in the athletic department.
Come see if your favorite school is a both a top football and basketball school!