I am so glad I live in the UK. Council tax is used to pay for libraries so joining and borrowing books is completely free at the point of service. Saying that my local council doesn't have a clue what e-books are and when there was a library consultation seemed to think that e-books meant lending out Kindles for people to then download things from Amazon *facepalm*, I gave up trying to educate them when I realised that the majority of others in the debate didn't really know what e-books were or how they are delivered by libraries either.
To answer your question, I wouldn't pay a fee unless I had to also pay a fee for printed material. At that point I would probably stop using the library as I believe it should be free to all at the point of service.