Radical idea, I know.
My public library took a picture of our library card alongside the book information. I have no idea how they sorted those records. I always imagined someone sitting in the basement going through rolls and rolls of microfilm in order to determine that I had, in fact, not returned Annie Oakley: Little Sure Shot.
In middle school, each book had a card, and students would fill out their name and homeroom. Books were filed by due date, and dutifully moved if a book were renewed. To find overdue books, we would type a list from the cards remaining in the oldest dates. There was no good way to check if someone had an overdue book except these typed lists, so they were very carefully kept.
No idea how they did it in high school. For reasons I can't even remember (probably something to do with the fact we could only come to the library from study hall 4 days a week-- I lived in the library in middle school), I was tremendously irritated by the high school librarians, so I spent four years borrowing books without checking them out. I returned every one of them. Such a rebel!
Struggles with Destiny? Taking too long to e mail those overdue notices home? Ha! Computers make things much easier, but I often wish that they didn't exist!