Speaking only for us, we do not require log books for the regular (tourist) fun diver as long as they have a c-card. We are rock-solid stict on depth standards so if you give us a Scuba Diver card, you only go to 12m. Period.
If you cannot produce a c-card, we can (if you're PADI certified) look it up online. If the info does not appear online, then you must produce a log book or you don't dive. Period.
We have no issues with computer logs, as long as they are the ones generated by a manufacturer's software (Suunto's Dive Manager, for example). My thinking is those are actually harder to forge than paper logs.
Now, we DO require dive logs in situations where you must meet specific prerequistes for a course. PADI Tec Deep, for example, requires X number of dives deeper than 30m, X number of nitrox dives, etc. We need proof of those before you can take the course.
Likewise, all our dive insturctor intens must produce logs. We had a couple interns who lost their books in Hurricane Katrina, but when they came to us we treated them as if they had 9 logged dives (they started as AOW). Of course, our internship program for themn is 4 months long, so they had plenty of time to get the 100 needed for the IDC (and much more).
Log books alone, however, won't necessarily get you on our boat. We had one customer show up with a collection of the rattiest log books you'll ever see. However, he admitted he was not certified, but he'd still done "500 dives" in his lifetime. He'd been having increasing problems getting to dive and now wanted to get his Open Water cert.
Using PADI's standard for "Experienced Divers" we arranged a shorter OW course for him only to discover that despite him saying he had 500 dives, he was, to put it kindly, crap in the water. Then we game him the exam and he failed.
He continued to insist he was too experienced to have to take a full Open Water course, so we kindly showed him the door. Sadly, I'm sure he probably found some other shop to probably accommodate him for some quick cash.