Civilization survived for ages without the internet, so I'm sure that we'll be able to find some way to keep people occupied while in space without it.
There's an ongoing/recurring experiment that NASA runs where a group of pretend-astronauts spends several months in a pretend Mars habitat with limited internet connection to Earth [1]. A podcast [2] was recently released where they go in-depth and interview the participants, and they touch on the whole "what am I supposed to do without the internet for 6 months?" issue a lot, but it never really seems to be that big of an issue. They do get bored, but most of the participants do stuff like learn a new language, study a textbook, or read a lot of novels. They also have a lot of "work" (like experiments etc) to keep them busy every day.
1: http://www.businessinsider.com/ap-mars-research-crew-emerges...