Current public librarian here.
As long as you tell us when there's damage and pay when something is too damaged to fix, we're good. We don't keep track of whose kid tears what, unless you're bringing back ruined books weekly or something like that.
If you find a book that's damaged before you check out, bring it to the librarian so we can pull it and either fix it or discard it. If you find it when you get home, then tell us when you bring it back. We err on the side of believing the patron as a rule, so there won't be a charge.
Thanks Winterlight, that's reassuring--weekly destruction is maybe the line of "too much"? We're definitely not there!
About the board books... the library has a very, very limited selection of those. And 2yo attacks books his older brother borrows too. So if people thought we were creating too much damage (which some people on here seem to be hinting at?), if he damaged another book and hasn't outgrown it like I'm hoping, then I'd just stay away from the library for the next 6 months or whatever until 2yo completely stopped tearing up books. We have lots of books at home (including board and a few cloth books). It's just really nice to get the variety that the library offers. And nice for them to go somewhere other than home.
Weekly destruction would be a "Time to have a talk with your child and also put the books where he can't get them," thing for me. The occasional torn book happens, if it was that often it would probably be more of a concern (for the parent having to buy replacements, at least!)
We used to tell our Aussie pup, "Gentle kisses are permitted," because she was something of a staplemouth when she got excited. So maybe some variation of that for books? It worked for Addie, at any rate, at least as far as her not biting us (much.)