Here in Denmark, smaller libraries still exist and have taken up the challenge of getting more readers, by housing reading clubs and making access easier.
We have the system mentioned from N. Ireland with an entry code, so you can go in at any time that suits you. As all books have for years been checked in and out electronically by the person borrowing them, this seems to work very well.
The public libraries are manned for differing numbers of hours weekly, depending on the demand. They house the counters of the Danish equivalent of Citizens' Advice in country areas, and even in larger cities. Here you can renew passports or driving license, apply for pensions and of course, receive advice in general. The libraries also house various voluntary schemes, such as local archives, historical societies and the nation-wide senior citizens' association runs computer classes and the like. During school holidays there are various book and computer related activities for children whose parents are not on holiday.
In smaller towns and villages, free legal advice is provided at the libraries on a fixed date every month. All these activities tend to bring people who would not usually go to a library in to them and most are curious enough to look around and perhaps borrow a book, cd, or dvd, while they are there.
As the public library catalogues have been on a homepage for years now, you can reserve books from your nearest public library or any other in the country over the computer and collect them when they are in. You can also access the various university libraries, the Royal Library, public archives etc. by computer, either from your home or from the nearest library. Books that research libraries do not permit you to take home, can be made available in the reading room of your nearest public library.
School children regularly are taken to visit the children's sections of the library and encouraged to read there, or take books home.
All public libraries are still funded by the state, and paid for by tax-payers through income tax, rates, etc.