I mean, who asked it first? Is it possible to at least date the question somehow?
WAG… Henny Youngman?
I heard it in a Bugs Bunny cartoon. The one where he’s supposed to be Groucho Marx. That would put it in the 50’s sometime.
No-one ever asked it. Someone used it as an example of a supremely loaded question. The question presupposes that you have beaten your wife. Ask someone (eg a politician running for office) such a loaded question, and then you can write a story in the newspaper along the lines of “Candidate denies beating wife”. Next thing, everyone thinks he did. Mud-Slinging 101.
Not quite …
“Candidate denies he has stopped beating his wife” or “Candidate confirms he has stopped beating his wife”. Either way, he has admitted beating his wife …
The question is found in print in US papers as early as 1901.
I believe that the asking of an unanswerable question (in that either answer you give makes you look bad) was discussed in ancient Greek dialogs. From either Plato or Aristotle, I think.
So it’s considerably older than either Henny Youngman or Bugs Bunny.
Beat my wife…Please. 
I credit and quote rjung for the following.
I asked a similar question in another thread, and here was his/her answer.
“Do you still beat your wife?”
“Mu.”
(Japanese Buddhist term meaning, “Your question cannot be answered because it is based on false assumptions.” Handy term to keep in your back pocket.)