Despite recent re-popularization of this tired trope by Jonathan Mayo and Emma Craigie in their book Hitler's Last Day: Minute by Minute, this is with certainty bordering on the absolute not true.

While I haven't seen yet what alleged evidence they dredged up for this, Hitler's childhood doctor, Eduard Bloch, had emigrated to the US in 1938 and had testified to American interrogators that everything was fine with Hitler's genitals as is described in Eduard Bloch: My Patient Hitler. In: Collier’s Weekly, March 15. and March 22. 1941; Eduard Bloch: The Autobiography of Obermedizinalrat Eduard Bloch. In: J. A. S. Grenville and Raphael Gross (Eds.): The Leo Baeck Institute Year Book, XLVII (2002); and Office of Strategic Services, Hitler Source Book, Interview With Dr. Eduard Bloch March 5, 1943.

While the story of one testicle is almost certainly attributable to the song "Hitler's only got one ball", the obsession with Hitler, his sexuality and his genitals goes back to the 1920s. In the Weimar Republic it was very common to start smear campaigns against your political opponent involving their sexuality. The Nazis and their newspaper "Der Stürmer" were experts in this, e.g. involving the President of Berlin's police force, a Social Democrat routinely smeared him as a homosexual. But this wasn't necessarily confined to the Nazis. If I recall correctly, the communist newspaper Die Rote Fahne also ran several articles on Strasser's homosexuality and even Hitler's sexuality as to denounce them.

Another thing to take into account is the popularity of Freudian psychology at the time. Explaining a person's behavior through the Freudian method heavily involves their sexuality and childhood experiences and this method was also used for Hitler by various people, including the American OSS during the war in order to gain a better understanding of him.

The third factor is the image Hitler projected of himself. According to Kershaw and others, Hitler's resistance to making his relationship with Eva Braun public among other things stems from the image of the "Führer" he wanted to project: A solitary and celibate fighter who was not concerned with the pleasures of every-day life but only with the German people. Several historians also pointed out that this was intended to make him more popular with women in nazi Germany because they thought they could have Hitler's children.

As for the post-war obsession, Freudian psychology and the great man view of history certainly play a role in the continued effort to find out things about Hitler's sex live. The quest to have an easy answer for the horrors of Nationalsocialism that is somehow to be found in Hitler's personality rather than in the study of the political and social climate of German at the time is one of the major driving forces behind this.

There also is a political dimension to the last factor mentioned: People like the authors of the debunked and terrible book "The Pink Swastika" made it their agenda to paint Hitler as a homosexual as to make a political statement against social acceptance of homosexuality.

Sources:

• Ian Kershaw: Hitler.

• Richard Evan's Third Reich Trilogy.

• Peter Longerich's Hitler biography.