I'm honestly surprised at most of these answers, as I thought there would be more responses of people being "online" in the 80s. I guess that makes me old!
Dialing up CompuServe or connecting to a BBS at 300 baud in 1982 was an interesting (and sometimes frustrating) experience. I still have my 300 baud Vicmodem and I think I have my Vic 20 somewhere. I definitely have my other Commodore hardware as well (see sig).
Well, I had a C64, but there wasn't much point in going "online" back then. Many games supported split-screen multiplayer so we just gathered around the computer and challenged each other in games like Lotus, International Karate etc. BBS's and the very early Internet were mostly viewed as something for perverts looking for pr0n (even more so than the current Internet), or something hackers used to discuss programming and computers.
What prompted me to get online was AmiNet. It was huge in the mid-late 90's..games, alternative operating systems, music, animations, tools and GUI enhancements, scene demos... Kind of like the huge repositories for Linux. There's still nothing like it for Windows.