Mark Anthony McDermott was born on October 26, 1961, in Waterbury, Connecticut, to Diane and Richard McDermott. He has a younger sister, Robin. Diane was 15 and Richard was 17 when McDermott was born; by 1967, the couple had divorced, and Diane and her two children were living with her mother, Avis Marino. On February 9, 1967, Diane died of what was believed for decades to be an accidental gunshot wound. In 2012 it was revealed that it was not an accident but that she was murdered by her then gangster boyfriend, John Sponza, who had already died by that time.
McDermott and his sister were raised by their maternal grandmother Avis in Waterbury. As a teenager, he began taking trips to visit his biological father, who owned the West Fourth Street Saloon in Greenwich Village, New York. McDermott worked in his father's bar, serving drinks and breaking up fights. He also fast-talked his way into the Mudd Club and Studio 54.
McDermott was uncomfortable with himself as a teenager, saying he had a "Dorothy Hamill hairdo". He began imitating his acting heroes, such as Marlon Brando and Humphrey Bogart, adopting their demeanor. In 1979, McDermott graduated from Holy Cross High School in Waterbury.
McDermott's father's third wife was playwright Eve Ensler, who adopted McDermott when he was 15 and she was 23. She later divorced his father. Ensler, with whom McDermott has remained close, encouraged him to pursue an acting career, and began writing roles for him into her plays. After Ensler suffered a miscarriage, he took the name Dylan, the name planned for her unborn child. He then graduated from Jesuit-run Fordham University with a BA in 1983, as well as studying under Sanford Meisner at the Neighborhood Playhouse School of the Theatre in New York City.