He is best known for his performance as John Merrick in the stage and television versions of The Elephant Man, a role for which he received a Best Actor nomination in the 1979 Tony Awards.
He also had a recurring guest role as the Bajoran priest Vedek Bareil on Star Trek: Deep Space Nine.
His feature film debut was in The All-American Boy (1973), and his first television appearance was in the PBS The Adams Chronicles (1976).
In 1979, while still an unknown, Anglim optioned the London play The Elephant Man, and debuted off-Broadway at St. Peter's Church.
In 1982, Anglim appeared in the ABC television version of The Elephant Man, and earned an Emmy nomination for Best Actor for the performance.