To get a public library card, you need to prove residence. In the past, you'd bring a utility bill with your name on it. But now, with most bills are paid online, how are you supposed to prove residence? Many libraries seem to be focusing more on borrowers proving state residence that city residence. But, when they're not, they accept mail, checkbooks, and leasing agreements, all of which seem easy to fake. To borrow physical books and periodicals, this is probably more trouble than it's worth, especially because you still need to pick the books up in-person. But, for the digital catalog, people who'd lack access otherwise might find it worthwhile.
Proving physical residence is important for accessing any local government service. So, while "outsiders" using the library is fairly innocuous, "outsiders" accessing other social programs may pose larger problems.