Clarke, who was born in Leicester, was educated at the University of Cambridge and Guy's Hospital, London, where he qualified in 1932. He remained at Guy's until 1936 when he engaged in life-insurance work before spending the war years in the Royal Navy. From 1946 Clarke worked as a consultant physician in Liverpool until 1958 when he joined the staff of the university. Here he later served as professor of medicine from 1965 to 1972 and also, from 1963 to 1972, as director of the Nuffield unit of medical genetics.

Although a consultant physician, Clarke was also a skilled amateur lepidopterist. In 1952 he became interested in the genetics of the wing colors of swallowtail butterflies and began a collaboration with Philip Sheppard, a professional geneticist who later became a colleague at Liverpool University. In particular, they worked on the inheritance of mimicry in the wing patterns of certain swallowtails. They noted that the gene controlling the wing pattern is actually a group of closely-linked genes behaving as a single unit – a supergene. They also found that even though the males also carry such supergenes, the patterns only show in the females.

At this point Clarke was struck by certain striking parallels between the inheritance of swallowtail wing patterns and human blood types. Above all it aroused his interest in Rhesus babies. This condition arises when an Rh-negative mother, that is someone whose blood lacks the Rh factor or antigen, and an Rh-positive father produce an Rh-positive child. Occasionally the fetus's blood leaks from the placenta into the mother's blood and stimulates the production of Rh antibodies. This will cause her to destroy unwittingly the red cells of any subsequent Rh-positive babies she may carry.

Clarke and Sheppard puzzled over how to prevent the mother producing the destructive Rh antibodies. The answer eventually came from Clarke's wife who in an inspired moment told him to inject the Rh-negative mothers with Rh-antibodies. As this is what destroys the blood of the fetus in the first place, the answer initially sounds absurd. However the Rh-antibodies should destroy incompatible Rh-positive cells before the mother's own antibody machinery acted, that is, before the mother could become sensitized to Rh-positive blood.

In 1964 Clarke and his colleagues were able to announce a major breakthrough in preventive medicine. Since then thousands of women have received injections of Rh-antibodies with only a few failures.