Stephenson was an All-American at the University of Alabama, under coach Bear Bryant. Bryant called Stephenson the best player he ever coached, regardless of position. He was drafted by Don Shula and the Dolphins in the second round of the 1980 draft. The 6'2", 255 pound center would eventually become regarded by many[who?] as one of the best centers of all time, but was used on special teams only until late in the 1981 season, when Mark Dennard was injured.
With the exceptionally explosive Stephenson as offensive captain, the Dolphins offensive line gave up the least sacks in the National Football League for a record 6 straight seasons, from 1982–1987, which doubled the length of the previous record. After he left, the Dolphins' line would continue to protect the pocket superbly, extending the record to 9 straight seasons.
On December 12, 1994 he was added to the The Miami Dolphin Honor Roll.
Despite the brevity of his career, in 1999, he was ranked number 84 on The Sporting News' list of the 100 Greatest Football Players. Also, in 1999, he was inducted into the Virginia Sports Hall of Fame, and in 2011 he was inducted into the Hampton Roads Sports Hall of Fame, for his contributions to sports in southeastern Virginia.