Dobkin began working in radio to pay for his studies at the Yale University School of Drama. He understudied on Broadway before serving with a radio propaganda unit of the United States Army Air Corps during World War II. When he returned to network radio he was one of five actors who played the detective Ellery Queen in The Adventures of Ellery Queen. In The New Adventures of Nero Wolfe (1950–1951), Dobkin played detective Archie Goodwin opposite Sydney Greenstreet's Nero Wolfe.
While playing Louie, The Saint's cab-driving sidekick on NBC Radio in 1951, he was asked to step into the lead role of Simon Templar to replace Tom Conway for a single episode — making Dobkin one of the few actors to portray Leslie Charteris' literary creation.
His other radio work included Escape (1947–1954), Gunsmoke (1952–1961), Yours Truly, Johnny Dollar (1956–1960), and the anthology series Lux Radio Theater. "The few of us who are left," Dobkin said of his radio days not long before he died, "keep telling each other that we never had it so good."He was also Lieutenant Matthews on The Adventures of Philip Marlowe, along with several other characters.
Continuing to work as a voice actor throughout his career, Dobkin contributed to the video game Tom Clancy's Rainbow Six: Rogue Spear (1999).