The first was actually in Nashville, TN, USA. Nashville Public Library opened up a reading room at the airport in 1962.
Amsterdam airport also has it's own library.
Although at both airports you can't actually take books out, so really, they're both more like reading rooms.
In Taiwan, just this year, they opened up the first airport ebook library.
"... four hundred titles in its collection, available in English or Chinese, and in ePub or Zinio formats, which can be read in the airport, but not taken away or downloaded onto the user's own device. Whilst publishing house Harper Collins has placed a limit on the number of times their ebooks can be downloaded in libraries, initiatives such as this imply that ebooks could become an integral part of library services in the future."
Nothing worse than being stuck at the airport without reading material, and then being short of cash or forced to buy just any paperback from the store. Airport libraries! woohoo!
What with libraries at risk in the UK, we know we need to support them by buying less and borrowing more.