Welch was born in Hollywood, California, into a show business family. His father, Robert L. Welch Sr., was a producer and screenwriter at Paramount Pictures, producing films starring Bob Hope and Bing Crosby. Welch Sr. produced the 25th Annual Academy Awards TV special in 1953 and "The Thin Man" TV series from 1958–59. Bob's mother, Templeton Fox, was a singer and actress who worked with Orson Welles' Mercury Theatre in Chicago and appeared in TV and movies from 1962 to 1979.