Robbins was born in Auburn, Alabama, the son of plant physiologist William Robbins. He obtained his MD from Harvard Medical School in 1940 and from 1942 to 1946 headed the virus and rickettsial section of the US army's 15th medical general laboratory. Here he worked on the isolation of the parasitic microorganisms causing Q fever, which are also responsible for certain kinds of typhus.

After World War II Robbins became assistant resident at the children's Hospital, Boston. In 1948 he became a National Research Fellow in virus diseases, working with John Enders and Thomas Weller. By 1952 Robbins and his coworkers had managed to propagate the poliomyelitis virus in tissue cultures. They established that the polio virus can multiply outside nerve tissue and, in fact, exists in the extraneural tissue of the body, only later attacking the lower section of the brain and parts of the spinal cord.

This research enabled the production of polio vaccines, the development of sophisticated diagnostic methods, and the isolation of new viruses. In recognition of this work, Robbins, together with Enders and Weller, received the Nobel Prize for physiology or medicine in 1954.

Robbins was director of the pediatrics and contagious diseases department at Cleveland Metropolitan General Hospital, and professor of pediatrics at the Case Western Reserve University, from 1952 until his retirement in 1980. He is married to Alice Havemeyer Northrop, daughter of the Nobel Prize winner John Northrop.