He's afraid of being debunked in public and have/having his reputation ruined.
The coordinating conjunction and connects two gerund-participial clauses: "being debunked in public" and "having his reputation ruined". Both are objects of the preposition of.
If we use have instead of having, we will violate the parallelism of this construction.
So yes, having seems the correct choice.
Your second question,
Is it common to use have in these types of sentences in informal speech?
..is very interesting. Maybe being and having clash in some way, and one feels the urge to use have? Maybe "being" and "having" make the construction feel non-parallel, and one instinctively tries to remodel the sentence to:
He's afraid of being debunked in public, and also afraid to have his reputation ruined.
..and this remodeling goes only part of the way in speech, leaving us with
He's afraid of being debunked in public and have his reputation ruined.
Could it be so? I'm not a native speaker, so I'm not sure. (0:
What if we left only being in the sentence?
He's afraid of being debunked in public and being left with his reputation in ruins.
I feel the urge to get rid of the second being here:
He's afraid of being debunked in public and left with his reputation in ruins.
Now the parallel structures are "debunked in public" and "left with his reputation in ruins", and they hinge not on of but on being. Maybe because of this ability of being to take on parallel structures one can feel that being and having clash with each other?
Just a conjecture.