Stucker was born in Des Moines, Iowa. His family moved to Shaker Heights, Ohio, where he distinguished himself in school as a pianist and class clown. He graduated from high school in 1965. Stucker made his screen debut co-starring in the 1975 comedic sexploitation film Carnal Madness as Bruce Wilson, a gay fashion designer who escapes from an insane asylum with two fellow inmates, fleeing to an all-girls school. He went on to perform in the 1977 earthquake-in-Los-Angeles comedy Cracking Up, alongside Fred Willard, Michael McKean, and Harry Shearer. In 1977 he appeared in the John Landis film The Kentucky Fried Movie, based on the troupe's sketches. This led to his supporting role in the Zucker-Abrahams-Zucker comedy Airplane!, which he reprised in Airplane II: The Sequel. For the initial film, the writers gave Stucker the straight lines for his scenes and let him write his character's off-the-wall responses. In 1982 he had a guest role in a three-episode sequence in the TV series Mork & Mindy and, in 1983, had a small featured role in Landis' Trading Places. In 1984, he had a co-starring role as the sex-obsessed psychiatrist, Dr. Bender, in the teen comedy film Bad Manners (aka: Growing Pains).

On July 12, 1984, Stucker was diagnosed with AIDS. He later publicly announced his illness, making him one of the first actors to announce he was suffering from the disease. Stucker had apparently suffered from many different types of cancer-related symptoms as early as 1979, prior to public knowledge of what AIDS was. He died from AIDS-related complications on April 13, 1986 at the age of 38. He is interred in the Chapel of the Chimes.