What? It's never been horrible. What do you mean "more acceptable"?
What the hell! Henry is fine. It's just an ordinary name, like James or something, but less popular and therefore more distinctive. I wouldn't blink at a little Henry (except to say "Oh, what a nice-named kid!") Especially since Henry's in the top 100 now.
No, it's not horrible in any way. It surprises me to even hear that question asked and kind of offends me that someone "thinks" it's becoming "more acceptable." (But only because I'm defensive about Henry.)