The cyberpunk era has begun. Sorta. We're not yet at the stage of having chips in our skulls, conducting all our business and personal interactions as avatars in a virtual world. Nor do we always have electronic information floating in front of our eyes as a heads-up display (although the first step of augmented reality is available for the public in the Netherlands). We're in an early transition stage where some of the information and connections are in cyberspace, some in the physical world. We have no choice at this point but to live in both worlds, as anyone who knows who has been told that a certain job can only be applied for online, or who wants to read an old book that has not yet been scanned into electronic form.
Libraries bridge both worlds. One obvious way is by providing both computer access and access to physical information (i.e., books, etc.). It's easier for people to explore a new world when the risk is removed -- when it's free of charge and in a supportive atmosphere. The thing is, many people live mostly in one world or the other. They don't see enough of the other world to know what they're missing. A bibliography in the back of a book isn't going to be of much help to someone who always goes straight to Google, nor a search engine to someone who only looks on bookshelves.
Libraries also provide "cross-cultural" guidebooks, maps, travelers' tips, and personal guides (i.e., librarians). A library might (as many have done) put its card catalog online, with an option to place holds remotely, maybe with a "lite" version of the website easily viewed on a smartphone's small screen. There might be tweets or a blog or e-mail notices giving updates of new books. Many libraries teach people how to navigate cyberspace with classes or simply with patient, nonjudgmental help from a friendly librarian. Flyers of new and interesting websites might be printed and laid on the counter next to the bestsellers list. The most astute libraries realize that it's not just a matter of providing the best print resources to those who prefer print, nor just a matter of providing the best online resources to those who spend their days online. It's also necessary to help both groups cross the border.