"Baby Doll" is a movie, based upon Tennessee Williams' one act play "27 Wagons Full of Cotton." It was directed by Elia Kazan and produced by both Kazan and Williams.
"A Streetcar Named Desire" is a play written by Tennessee Williams. The filmatic adaptation of the play was produced by Elia Kazan.
Therefore, one finds the first similarity between the movie and the play: Tennessee Williams and Elia Kazan worked on filmatic versions of both.
Secondly, both pushed the bar when it came to sexuality. In "Baby Doll," the 19 year old Baby Doll (the name of the character) is overly sexual. In the same way, one can define Blanche du Bois, from "A Streetcar Named Desire." The ending rape of Blanche by Stanley shocked both readers and viewers of the play, while the movie shocks viewers with Baby Doll's blatant sexual nature (even being deemed "possibly the dirtiest American-made motion picture that has ever been legally exhibited" by Time Magazine).
Essentially, both the play and the film deal with issues the public was not ready for. The blatant sexual nature of both Baby Doll and Blanche, the rape of Blanche, and the marriage of a much older man to a much younger (virginal "child) proved to be more than audiences of the time could handle.