From 1948-62, the family lived in West Germany, before moving back to the United States. After studying at Clongowes Wood College (a Jesuit school in Ireland) from 1955–61, he attended the University of Tübingen (Walsh spoke fluent German), and then the University of Rhode Island, where he starred in many college theater productions. In 1974, he was discovered by a theatre Director and began working in off-Broadway shows. After college Walsh worked briefly as a VISTA volunteer in Newport, Rhode Island organizing tenants for the United Tenant Organizations of Rhode Island (UTO) before resigning to pursue his acting career.

Walsh did not appear in films until 1983, when he had a minor role in Eddie Macon's Run. Over the next 15 years, he appeared in over 50 feature films, increasingly taking the bad guy role for which he is well known, such as Sergeant Major Dickerson in Good Morning, Vietnam. On television, he again portrayed an evil character, prison warden Brodeur on The X-Files in 1995 in the episode "The List".

Walsh wanted to show his range as an actor and play good guys, despite being typecast as a villain. He played relatively decent characters in Outbreak and Sniper, and also played the rather sympathetic Marine Lieutenant Colonel Matthew Markinson in A Few Good Men. He played a member of Majestic 12 in the 1996 sci-fi drama series Dark Skies. The 1997 thriller Breakdown featured Walsh as the villainous truck driver. It was his last starring film released during his lifetime. In his final year of life, Walsh starred in Hidden Agenda, Pleasantville, and The Negotiator. All three films were dedicated to his memory.

Walsh died of a heart attack in La Mesa, California, at Grossmont Hospital on February 27, 1998, at age 54, after feeling ill and collapsing while a guest at the Optimum Health Institute in Lemon Grove, California. Jack Nicholson dedicated his Academy Award for As Good as It Gets to Walsh's memory. The two had acted together in two films, A Few Good Men and Hoffa.