Kulp was born in Harrisburg, Pennsylvania, the only child of Robert I. Kulp, a traveling salesman, and his wife, Marjorie S. Kulp. The family moved from Mifflintown, Pennsylvania, to Dade County, Florida, prior to 1935.

Kulp obtained her bachelor's degree in journalism from Florida State University in 1943, worked on a master's degree in English and French at the University of Miami, and was a member of Pi Beta Phi Women's Fraternity. In the early 1940s she also was a feature writer for the Miami Beach Tropics newspaper, writing profiles of celebrities such as the Duke and Duchess of Windsor and Clark Gable. In 1944 Kulp left the University of Miami to volunteer for United States Naval Reserve service in World War II. As a member of the WAVES (Women Accepted for Volunteer Emergency Service), Lt. J. G. Kulp received several decorations, including the American Campaign Medal, the National Defense Medal, and the Good Conduct Medal. She left the service in 1946.

Kulp married Charles Malcolm Dacus on April 1, 1951 in Dade County, Florida; they divorced in 1961.

Late in life Kulp indicated to author Boze Hadley in a 1989 interview that she was lesbian. "As long as you reproduce my reply word for word, and the question, you may use it," she said. "I'd appreciate it if you'd let me phrase the question. There is more than one way. Here's how I would ask it: 'Do you think that opposites attract?' My own reply would be that I'm the other sort – I find that birds of a feather flock together. That answers your question."

Shortly after her marriage, Kulp moved to Hollywood, California to work in a studio publicity department, but director George Cukor convinced her that she should work in front of the camera.

Her film debut as a character actress was in 1951 in The Model and the Marriage Broker. She appeared in subsequent films, including Shane, Sabrina, and A Star is Born. After working in television on The Bob Cummings Show, she returned to movies in Forever, Darling, The Three Faces of Eve, The Parent Trap and The Aristocats.

A reviewer once described Kulp as possessing the "face of a shriveled balloon, the figure of a string of spaghetti and the voice of a bullfrog in mating season." She was also described as television's homeliest girl.

In 1955 she joined the cast of The Bob Cummings Show (aka Love That Bob) with Bob Cummings, portraying pith-helmeted neighborhood bird-watcher Pamela Livingstone.

Kulp appeared in one episode of I Love Lucy. In the 1957 episode "Lucy meets the Queen," she showed Lucy and Ethel how to properly curtsy. She also appeared in several episodes of Perry Mason, The Jack Benny Program., 87th Precinct, The Twilight Zone and The Outlaws.

In 1962, she landed her breakout role of Jane Hathaway, the love-starved bird-watching perennial spinster, on The Beverly Hillbillies television series. She remained with the show until its cancellation in 1971. In 1967, she received an Emmy Award nomination for her role.