Arlene was born in Chicago in 1936 of Polish descent. She began her acting career in her early teens and went professional in a summer-stock troupe. After studying at the prestigious Goodman Theatre in Chicago, she made her way to New York City, where she studied with Lee Strasberg, Sanford Meisner and Uta Hagen.
A life member of The Actors Studio, She appeared in her first major production, The Night Circus with Ben Gazzara, at the Shubert Theater in New Haven, Connecticut on November 17, 1958. After a week-long trial run, the play moved to Broadway on December 2, 1958, but closed after only seven performances.
Despite that setback, she continued working in other plays such as Take Me Along with Jackie Gleason, Walter Pidgeon and Robert Morse (448 performances from late 1959 to late 1960), Neil Simon's first Broadway play, Come Blow Your Horn, which ran 677 performances from February 1961 until October 1962, and One Flew Over the Cuckoo's Nest, starring Kirk Douglas, from November 1963 until January 1964. Golonka appeared in two other Broadway plays from 1965-66, and took supporting roles in films produced in the New York City area.
Golonka had supporting roles in some 30 films, including Harvey Middleman, Fireman (1965), Penelope (1966), The Busy Body (1967), Welcome to Hard Times (1967), Hang 'Em High (1968), The Elevator (1974), Airport '77 (1977), The In-Laws (1979), Love At First Bite (1980), The Last Married Couple in America (1980), My Tutor (1983), The End of Innocence (1990) and A Family Affair (2001). Later in her career, she began teaching acting.
In 1992, Golonka appeared as Sally Nash in the 13th episode "Fool for Love" in season 5 of the television series In the Heat of the Night with Carroll O'Connor. In this episode she played the other woman to a philandering Dr. Vance Talbot (played by actor Robert Ginty) who tries to frame her for the murder of his wife before murdering Nash as well. Golonka was a regular on the animated cartoon Speed Buggy, providing the voice of "Debbie", and had a recurring role on the short-lived TV series Joe & Valerie. She performed voices in other animated series including The New Yogi Bear Show, Capitol Critters, Yogi's Treasure Hunt and The New Scooby-Doo Movies.