Dennis was the son of Bing Crosby and his first wife Dixie Lee. With his twin brother Phillip Crosby and his other brothers, he attended Bellarmine College Preparatory in San Jose. He graduated in 1952 and enrolled at Washington State University at Pullman.
In the late 1950s, Dennis and his brothers, Lindsay and Phillip, often performed as the Crosby Brothers in nightclubs and on the Ed Sullivan Show, though Dennis Crosby reputedly wished to avoid such appearances.
While a young man, Crosby performed military service in Germany.
On May 4, 1958, Crosby married Pat Sheehan, a Las Vegas showgirl and model who had once dated his father. She was also Miss San Francisco of 1950, Playmate of the month of October 1958, and part time actress. Within days, Crosby was sued by another woman, Marilyn Miller Scott, over the paternity of her daughter, Denise Crosby. The sensational lawsuit lasted three years and ended with Dennis being ordered to pay Scott child support and legal fees. This and the marriage to Sheehan and other details caused deep embarrassment for both him and his famous father. Although Bing died when his granddaughter was 19, the two reportedly never met.
Crosby and Sheehan had three sons: Gregory, Dennis Michael, Jr., and Patrick Anthony. In 1963, while working in Los Angeles for Bing Crosby Productions, he met Arlene Newman. On July 3, 1964, Crosby and Sheehan were divorced. Later that year, Crosby married Newman, with whom he had three daughters, including Kelley Lee Crosby and Erin Colleen Crosby. Additionally, Crosby adopted Newman's daughter, Kathy, from her first marriage to boxboy Mike Buelle, which ended in divorce. Dennis and Arlene divorced in 1991.
On December 26, 1989, Crosby's younger brother Lindsay committed suicide by a shotgun blast to the head. Deeply distraught by his brother's passing and his recent divorce and grappling with alcoholism, Dennis Crosby committed suicide on May 7, 1991 in Novato, California, also with a shotgun. On January 14, 2006, Dennis's former wife, Pat Sheehan, died at the age of 74.