Born in London, Robbins was a bank clerk who became an actor after appearing in amateur dramatic performances in Hitchin, Hertfordshire, where he and his family lived at the time. Robbins made his television debut as the cockney soldier in Roll-on Bloomin' Death. Primarily a comedy actor, he is best remembered for the role of Arthur Rudge, the persistently sarcastic husband of Olive (Anna Karen), in the popular sitcom On the Buses (1969–73). Robbins and Karen provided the secondary comic storyline to Reg Varney's comedy capers at the bus depot. Robbins also appeared in the series film spin-offs, On the Buses, Mutiny on the Buses, and Holiday on the Buses. His other comedy credits include non-recurring roles in Man About the House, Oh Brother!, The Good Life, One Foot in the Grave, The New Statesman, George and Mildred, Hi-de-Hi! and You Rang, M'Lord? He appeared as a rather humorously portrayed police sergeant in the TV adaptation of Brendon Chase.

As well as these comic roles, he assumed various straight roles in some of the major British television shows of the 1960s and 1970s: including Minder, The Sweeney, Z-Cars, Return of the Saint, Murder Most English, The Avengers, Dixon of Dock Green, The Bill and the 1982 Doctor Who story The Visitation.