He studied at Harvard University (Bachelor of Arts, 1949) and University of California at Berkeley under John L. Kelley (Doctor of Philosophy on the thesis Structure of Operator algebras, 1955), while working as numerical analyst at Lawrence Radiation Laboratory (1953-1955).
He then joined Duke University (instructor, 1955-1956) and Massachusetts Institute of Technology (instructor, 1956). He served as a research associate at Harvard University (1973) and at University of California (1974 and 1980) while maintaining his professorship at Dartmouth College where he taught until his death in 1996. Prosser was also a cellist and sailing devotee.
The mathematics department at Dartmouth College opened the Prosser Memorial Lectures in 2002.
