His death was confirmed by his son, Daniel.
Elkins was a colorful figure who entered show business in 1950 as an office boy at the William Morris talent agency in New York City. He soon had his own management company and eventually handled the careers of Steve McQueen, Sammy Davis Jr., Robert Culp and James Coburn.
One of his earliest producing successes was "Golden Boy," a 1964 musical adaptation of the tragic 1938 Clifford Odets play about a reluctant boxer, which starred Davis.
Elkins' biggest hit was "Oh! Calcutta!" (1969), a musical comedy that reflected the exuberance of the 1960s with its unabashed nudity and liberal use of four-letter words. The show was panned by critics, who could not find a plot amid the cast's constant stripping and dancing. "Voyeurs of the city, unite, you have nothing to lose but your brains," New York Times theater critic Clive Barnes wrote in 1969.
But Elkins did not mind the criticism.
"There is nothing you can say about 'Oh! Calcutta!' that doesn't sell 10 tickets," he told The Times a few months into what would be a 20-year run on Broadway that ended in 1989 after more than 7,200 performances. It was made into a 1972 film, which Elkins co-produced with others.
The same year "Calcutta" made its controversial debut, Elkins produced "Alice's Restaurant," a movie directed by Arthur Penn and based on the semiautobiographical antiwar song by Arlo Guthrie. Elkins also produced the 1971 comedy "A New Leaf," starring Elaine May and Walter Matthau.
His other Broadway credits include "The Rothschilds" (1970), "An Evening with Richard Nixon and …" (1972) and "Sizwe Banzi Is Dead" (1974).
Elkins was born in New York City on Oct. 18, 1929. While studying drama at Brooklyn College, he opened a tent theater in Queens with his friend Gordon Davidson, who would go on to become founding artistic director of the Center Theatre Group at the Music Center of Los Angeles County.
Elkins gave up plans to attend law school after he went to work at William Morris. After a short stint in the Army making training films during the Korean War, he started his own agency.
By the late 1960s, he was living in a Manhattan penthouse, where he kept an office dominated by a huge portrait of Napoleon. "People walk in and look at me and look at him and register my psychotic need," Elkins, a short man with a neatly trimmed goatee, said jokingly in a 1969 interview with The Times.
He had six marriages, including one to actress Claire Bloom. She starred on Broadway in the Elkins-produced revivals of Ibsen's "A Doll's House" and "Hedda Gabler" in 1972. Married to actor Rod Steiger when she and Elkins began an affair, she described the producer in her 1996 memoir "Leaving a Doll's House" as a "buccaneer" who aggressively pursued her until she agreed to marry him in 1969. Their marriage dissolved in 1974 when he ran off with another woman.
"The only thing that saved me during my marriage to Elkins, and I do not in any way belittle it, was that Elkins was a fine theatrical producer," she wrote.
Elkins moved to Los Angeles in the 1970s. His other survivors include his sixth wife, Sandi Love; another son, Johnny; and a grandchild.