I agree that the connection between Hoover and The Crucible is tenuous; without any historical or biographical context for Arthur Miller and his writing, Hoover is a non-entity to this play. However, because literature is not written in a vacuum and because the author had the clear intention to condemn the anti-Communist "witch hunt," Hoover is connected--and condemned.
The Crucible opened on Broadway in 1953, right in the thick of the McCarthy Era where hearings were being held in the House and Senate, and a large number of actors and playwrights had been called to testify about communist activities and memberships. Arthur Miller, author of The Crucible, published the work as a criticism of McCarthy as it was happening. The FBI had long suspected he had communist ties, and for a time he may have been under surveillance at J. Edgar Hoover's orders. It's also believed that Hoover himself ordered that Miller be denied a passport to attend The Crucible's opening in London.
Miller in 1956 married Marilyn Monroe, and was asked to testify before the House Un-American Activities Committee (HUAC), where he refused to name names. Monroe couldn't stand Hoover and Richard Nixon for the way they hounded Miller, and Hoover ordered her followed for a time too.
So the connection is that the play was a direct response to anti-communist McCarthyism in the 1950s, and Hoover was every bit the rabid anti-communist that McCarthy and Nixon were.
Another connection you could note of the Hoover presidency as it relates to The Crucible, might be the way he handled McCarthy and the investigations.
Danforth just blindly believed the girls. He allowed spectral evidence (which is listening to things not seen or easy to demonstrate as fact, like ghosts, visions, or dreams) into the courtroom. Danforth looked at illogical evidence and believed it to be true.
I cannot help but believe that Miller was suggesting this about Hoover by writing the play when he did. Most literature is a comment on society in one way or another and Miller was upset by what he saw in the Red Scare. People (including officials like Hoover) because irrational. Hoover had to have made some decisions that were illogical about people just because there was a thought that they might have been a spy for the Russians.
The clearest connection I can see between Hoover and this play is the fact that Hoover was involved in seeking out communists during the Red Scare/McCarthy Era in the United States. Since the play is meant to comment on this Red Scare, Hoover is connected to it. You might say that he is something like Danforth or one of the other characters who truly believes in the witch trials and wants them to continue.
As one example of what Hoover did, he kept files on all sorts of people who were suspected of being criminals. He then leaked information from those files to people connected to McCarthy and HUAC. The information was then used to help persecute the suspected communists.