Cohen was educated in his native New York, at the City College and at Columbia, where he obtained his PhD in 1941. He joined the University of Pennsylvania in 1943, serving as professor of biochemistry from 1954 until 1971, when he moved to the University of Denver, Colorado, as professor of microbiology. Cohen returned to New York in 1976 to take the chair of pharmaceutical sciences at the State University, Stony Brook, becoming emeritus professor in 1985.

In 1946 Cohen began a series of studies in molecular biology using the technique of radioactive labeling. The common microorganism Escherichia coli could be infected in the laboratory with the bacteriophage known as T2. Within a matter of minutes the bacterial cell would burst releasing several hundred replicas of the invading T2. The problem was to understand the process. It was known that phages were nucleoproteins consisting of a protein coat surrounding a mass of nucleic acid (DNA in the case of T2). But, as Cohen realized, nucleic acid differed from protein in containing measurable amounts of phosphorus. This could in theory be traced through any biochemical reaction by labeling it with the radioactive isotope phosphorus–32.

Cohen used this technique in a number of experiments in the late 1940s that suggested rather than demonstrated the vital role of DNA in heredity. It was not until 1952, when Alfred Hershey and Martha Chase used Cohen's labeling technique, that more substantial results were available.