After graduating from Cretin-Derham Hall in Saint Paul, Minnesota, Walsh attended and played college football with the University of MiamiHurricanes. After red-shirting a season and backing up Vinny Testaverde in his second, Walsh posted a record of 23–1 in his two seasons as the team's starting quarterback, losing only to Notre Dame, and led the Hurricanes to the 1987 national championship as a sophomore. The following season, Walsh finished 4th in the Heisman Trophy balloting. A pro-sized passer, he held the school record for career touchdown passes from 1988 to 2002 before being surpassed by Ken Dorsey. Skipping his final year of eligibility, Walsh left Miami in April 1989 and declared himself eligible for that year's NFL supplemental draft.
Selected by the Dallas Cowboys with the first overall pick in the 1989 NFL Supplemental Draft, Walsh was reunited with Jimmy Johnson who had been Walsh's head coach in Miami and had just taken over the Cowboys' reins. However, Dallas had also selected UCLA quarterback, and future NFL Hall of Fame enshrinee, Troy Aikman first overall in the regular 1989 NFL draft. Despite starting five games (and picking up Dallas' only win of the season, a 13–3 victory over the rival Washington Redskins) and playing in three other games, Walsh was never able to move out of Aikman's shadow. Four games into the 1990 season, Dallas traded Walsh to the New Orleans Saints for first, third and second round draft choices. With the third pick the Cowboys would eventually select Erik Williams.
Walsh led the Saints that season to their second ever playoff appearance finishing the season, at 8–8 and knocking the Cowboys out of the final playoff spot. He went on to have an 11-year NFL career, playing not only for the Cowboys and Saints, but also the Chicago Bears, St. Louis Rams, Tampa Bay Buccaneers, and Indianapolis Colts. After leading the Bears to a playoff victory and finishing the season with a 9–4 starting record, he went on to start just four more games over his next five years. He served as a backup to Trent Dilfer in Tampa Bay and Peyton Manning at Indianapolis. Walsh retired after the 1999 season with the Colts and finished his career with 713 completions in 1317 attempts (54.1%), 7,875 yards, 40 touchdowns, and 50 interceptions, for a passer rating of 66.4.
In December 2008, he accepted the head football coaching job at Cardinal Newman High School in West Palm Beach, Florida. After a 4–6 record in his first season, Cardinal Newman finished 7–4 in 2010 and earned its first playoff appearance since 2005. Walsh coached the team to 10-2 records in 2011 and 2012, before losing in the opening round of the 3A playoffs. In 2013, Walsh led Newman to their First District Championship in 10 years and the Regional Final. It was the school's fifth straight playoff appearance. In April of 2015, Walsh accepted the Director of Football position at IMG Academy, replacing Chris Weinke.