have I been asked for my C-card. I have been asked for my log book even less.
On my trips to the Flower Gardens, the boats (MV FLING and MV SPREE) always ask to see C-cards. They also request that a log book be available showing all of the dives taken within the last six months. (they require recent open water experience). However, I don't recall ever actually showing them my logs, but only that I had to sign an affidavit attesting to my recent experience.
I just got back from two weeks in Panama City, FL. I was not asked for my C-card on any occasion, although, oddly enough, I witnessed several other divers being asked for theirs (just lucky, I guess).
If I were the owner of a LDS or dive boat operation, I would REQUIRE that all divers present present a C-card and the ability to demonstrate minimal diving competency. This could be in the form of a logbook (admittedly forgeable), or perhaps, a shallow competency dive.
On most boat trips I've taken, outside of the Gardens, at least one or more divers on the boat exhibited such poor skills that I wondered if they were ever actually certifified.