By 1988 you had interesting collaborations over the internet using Xwindows and CMU's various Andrew explorations. IIRC it was too slow to do remote computing but you could load binaries over AFS from, say, MIT's network to run on a system at CMU (and vice versa). Most consumer networking, if you could call it that, was over the proprietary Compuserve, AOL, services as well as over Fidonet, Usenet (using uucp for point-to-point and nntp for networked servers). Colleges and universities also had BITNET, utilizing IBM's RSCS over SNA (mostly text, mostly IBM mainframes but not exclusively, for point to point email or mailing lists).
Web browsers hadn't been invented yet so the Internet in 1988 was mainly used at research universities and commercial labs. Definitely not a consumer product, but well on its way to being an important thing.
It's hard to remember the personal impact of this even having lived through it. I think a lot of homes (esp. those with children and/or adult in teaching, design, or business) had computers, but even by late 1988, most probably did not connect to any network, and nearly all of those that did connect somewhere did only to various BBS and services like The Source, CompuServe and AOL. There were viruses transmitted by floppy for some time and it was a minor headache, but I do not recall if in 1988 The Source et al. had an Internet portal. I think the content of the publicly available 2400 baud Internet then (which for a private individual would be accessing newsgroups through a service on dialup) then was probably text, maybe a lot of slow steamy pictures, email, demos and pirated games. I suppose the DoD, NSF, and similar may have been heavily using the Internet then in interesting ways beyond email and ftp servers, but I can't think of what else in 1988 they might have been doing on the Internet (other than getting pwned, of course).