The Darby Free Library in Darby Borough, Pennsylavania which claims to be the oldest public library in continuous existence is, by recent accounts,in jeapordy of closing if it doesn't receive additional financial support. While I sympathaize with the plight of this historic library, I would challenge its claim of being the oldest public library.
The Darby Library Company was founded by a group of Quakers in 1743 just 12 years after the founding of the Library Company of Philadelphia (which also remains in existence), the nations's first subscription lending library. From its founding up to an unknown point, the Darby Library, was supported by paid subscriptions from the members of the library and could only be used by its
members. It was only at the point that it became free to use by anyone in Darby Borough that it met the first criteria for a library to be considered a public library. It was not until it started receiving public tax support for serving the Darby Borough that it met the second major criteria for being considered a public library. This was also at some unknown point, but probably not until the 1990s. So for much of its history the Darby Library was a subscription library and therefore cannot legitimately claim to be the oldest public library.
The Peterborough Town Library in Peterborough, New Hampshire is the oldest public library in continuous exhistence to meet the two conditions of being free to the residents of the community it serves and to also receive public funding.
The Boston Public Library which also sometimes claims to be the oldest public library is the oldest city or large public library.
Of course, the Darby Free Library remains a unique and historic library and hopefully will receive adequate funding from Darby Borough.
For more on the early history of the Darby Library check out pages 219 to 222 in the 1918 volume of the Pennsylvania Magazine of History and Biography which can be found in Google books at this location on the Web.
The Straight Dope website does a good job of answering the question "How Did Public Libraries Get Started?"