Drummond was born as Alice E. Ruyter in Pawtucket, Rhode Island, daughter of Sarah Irene (née Alker), a secretary, and Arthur Ruyter, an auto mechanic. She graduated from Pembroke College (Brown University) in 1950.
She played Nurse Jackson on the TV series Dark Shadows. She was later nominated for a 1970 Tony Award as Best Supporting or Featured Actress (Dramatic) for her performance as Mrs. Lee in The Chinese and Dr. Fish. Miss Drummond was a regular on the CBS soap opera, "Where
the Heart Is," where she created the role of "Loretta Jardin," which she played until the series ended in 1973. She also appeared in a short-term role on another CBS soap opera, "As the World
She has appeared on television in guest roles on Kate & Allie, Boston Legal and Yes, Dear.
One of Drummond's notable movie roles is the librarian frightened by an evil spirit at the beginning of Ghostbusters. She also served as the senior citizen stereotype in the movie Furry Vengeance.