That's really difficult. You are getting in before the rise of the ISPs, but not much before. Let's see. Bob Rieger founded Netcom Online Communications in 1988, so you can't block that. Even if you could, there were enough people thinking in terms of dialup Internet access that it'd be hard to screw it up by blocking any one of them. The World out in the Boston area could have taken Netcom's place. PSINet probably could have managed it as well.
If you'll give me latitude of two months, hm. OK, best I can do:
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On November 2nd, 1988, Robert Morris accidentally released the Internet worm. He had intended it to merely measure the size of the Internet, but he made an error in his coding. His plan was for the worm to check potential targets to see if the worm had already infected them; if he got a "yes" in reply, 1 out of 7 times he'd do the infection anyhow. That would prevent people from blocking the worm simply by writing a program to answer yes to worm queries. If he'd been a better coder, everything would have been fine. But he made the classic careless mistake; he used "=" to compare two values instead of "==". As a result, the worm would always attempt to infect new targets no matter what the reply. This meant that any infected computer would quickly run out of system resources and become unusable.
The fledgling Internet ground to a halt. Most of the activity on the Internet was academic in nature, but the military was minorly affected and deeply concerned. The court of public opinion was quick to decide that the Internet could not be a Wild West any more.
Jon Postel had been slated to be Director of the new Internet Assigned Numbers Authority in December of 1989. That appointment was put on hold while Congress investigated the Morris Worm. The IANA was meant to have authority over the distribution of IP addresses and the naming scheme of the Internet. We can only guess what might have become of a non-governmental body with such authority.
The public's paranoia was cemented in April 1989, when Clifford Stoll's book The Cuckoo's Egg became a runaway best-seller. His story of how he chased a Russian hacker through the wilds of the Internet was gripping, interesting, and all too threatening to a public that was primed to mistrust the Internet. If the people attacking weren't just Americans, like Robert Morris, but also Russians? Too dangerous for words.
In July 1989, a bipartisan panel led by Senator Al Gore persuaded Charles M. Herzfeld, founder of ARPANET, to return to government service as the Director of a new organization: the Defense Advanced Internet Projects Agency (DAIPA). This agency was spun out of the Defense Advanced Research Projects Agency to manage the Internet before it became unmanageable.
Under Charles Herzfeld, by harsh necessity, use of the Internet remained restricted. In the late 1980s, Internet usage by college students was spreading -- but they were rarely justified users given the Internet's purpose. This ended by 1990.
And a good thing too. If the Internet had become a general purpose communication network, we would never have seen Fidonet rise to become the most thoughtful, contemplative means of communication ever invented. The inherent "store and forward" design of Fidonet, in which messages are collected and stored before transmission to the next node, permitted the development of the "really REALLY?" filter. Life without it seems unimaginable; consider what a mess email would be if it was sent as soon as you hit return? Many friendships are saved by the chance to think twice before a message is sent.
Further, free communication is not valued. When we must pay to call BBSes in other states, we value the time we spend on them and we are more careful in our words. It's basic marketing: free services aren't valued as highly.
In a very real way, Mr. Morris did us all a favor.
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Notes:
The = vs. == bug is the POD. In OTL, the worm did reinfect 1 out of every 7 computers, which turned out to be way too high a ratio anyhow... but it wasn't as bad as it could have been.
I don't know the exact date Stoll's book was published, but it was 1989. His original article on the hacker came out in 1988; I assume that Doubleday put more promotional muscle behind it and urged him to finish it more quickly given the increased attention given to the Morris Worm. It's also possible that they got a ghostwriter to help, making the book a bit more accessible to the masses, which would also help popularity.
After the big changes -- the Worm, the IANA going fully government, and Cuckoo's Egg providing political cover -- I didn't bother working anything out in detail. Fidonet probably would have been a bigger deal; it was a functional BBS network and there would have been an alternative. I'm pretending that AOL and Compuserve die to Fido the same way they did to the Internet in OTL, but you could make a case that AOL would be the dominant online service, probably.
If you wanted to be cute with this timeline, you've got every excuse in the world to hand the 2000 election to Al Gore. ITL, his assertions about his role in the early Internet would have been accepted, because the Internet never became the commercialized free zone it was in OTL. A minor thing but it helps shift the narrative, and we all know how close 2000 was.