At the age of eighteen Brodax enlisted in the US Army and served in World War II. He was wounded in action, and was subsequently awarded the Purple Heart, the Combat Medic Badge, and three battle stars.
From 1950 to 1960, Brodax worked in program development for the William Morris Agency, where he helped develop Your Show of Shows, Pulitzer Prize Playhouse, and Omnibus. He joined King Features Syndicate in 1960 as the head of their film and television development department.
After Paramount Studio's contract to produce Popeye cartoons ran out in 1957, King Features acquired the television rights. Brodax oversaw the production of over 200 new shorts. The rapid pace of production, coupled with low budgets, resulted in shorts that are held in low esteem by many Popeye enthusiasts. Brodax was also the producer of King Feature's animated revival of Krazy Kat, as well as Cool McCool, Beetle Bailey, and Casper the Friendly Ghost.
After seeing The Beatles on The Ed Sullivan Show, Brodax approached the band's management with the idea of producing an animated series featuring the Fab Four. The series, of which 40 episodes were produced, premiered on September 25, 1965 on ABC. Brodax was later involved in the production of the animated Beatles film, Yellow Submarine, as producer and co-screenwriter.
From 1969 to 1980 Brodax worked as a freelance producer/writer/lyricist/director. He was the animation supervisor for ABC's children's show Make a Wish (1970-1975), and for Animals, Animals, Animals (1975-1980).
In 2004, Brodax released a memoir, Up Periscope Yellow: The Making of the Beatles' Yellow Submarine. He currently resides in Weston, Connecticut, where he is the head of the Brodax Film Group, a television and production company.