Budde attended the University of Southern California, and played for the USC Trojans football team from 1976 to 1979. He was the first player since World War II to start as a freshman. Budde is one of USC’s most highly decorated offensive lineman. As a senior in 1979, he was a unanimous first-team All-American, runner-up in the Outland Trophy voting, USC Offensive Player of the Year, USC Most Inspirational Player, and an Academic All-American. He was also selected as the first and only Lombardi Award winner in USC's history. In 1980, Budde also earned the NCAA Post Graduate Scholarship. During his career at USC, Budde started in three Rose Bowls; each time the USC Trojans were Rose Bowl winners. In 1978, led by Coach John Robinson, their Rose Bowl victory earned USC a National Championship.
Budde again made history as the eleventh pick in the first round of the 1980 NFL Draft by the Kansas City Chiefs. Budde and his father, All-ProEd Budde, became the first and only father and son in NFL history to be drafted in the first round to the same team and play the same position. He played for the Chiefs through 1987.
On December 30, 2010, Budde will be inducted into the Rose Bowl Hall of Fame in a ceremony at the Pasadena Convention Center.
Off the field, Budde and his wife, Nicolette, devoted themselves to working with abused and neglected children through Camp Opportunity and Division of Family Services in Kansas City, Missouri.
Following retirement from the NFL in 1988, Budde returned to college and earned his Master’s in Physical Therapy from Loma Linda University. For the last 17 years, Brad has rehabilitated senior citizens as President of Budde Physical Therapy, Inc. From 1995–2005, Brad also worked as the offensive line coach at San Clemente High School and Orange Coast College.
In 2005 Budde founded GameDay Performance Systems, bringing the fundamentals of high performing sports teams into the workplace.
Budde lives with his wife Nicolette in Capistrano Beach, CA; they are the parents of two children, Sasha and Beau. Sasha is a high school English teacher in Irvine, CA, and won the Teacher of Excellence award for the 2011-2012 school year; she resides with her husband Sean in San Clemente, CA. Beau is a high school social science teacher in Irvine, CA; he and his wife Annette live in Orange, CA.