Alaska local state law, and maybe city ordinances, would ultimately bear upon this. I will try to add Alaska state to the topics. Without knowledge of the specific law, it's a guessing game.
I have a Mat-Su Borough library card that includes the Palmer library. I couldn't tell you what the local laws say about this. My feeling is that the library is free and public, and if the network is unsecured, it's his to use. I'm basing this on the library being being a tax-payer supported public service.
Whether or not it is unethical or illegal to tap into a private wireless network is another question.