Consider that arcade halls and LAN parties are very similar to each other. In both, you have a a group of people getting together in one place for the purpose of playing computer games.
The big difference, as I understand it, is that arcade halls were basically set up with single-player games. Because the technology wasn't very far advanced at the time, the games were also, by modern standards, relatively simple. But there is no reason why it has to be that way.
I can think of two ways arcade halls could remain a desirable element from a gamers' perspective:
Access to some types of games is restricted.
Maybe the government feels much more strongly than in our world about games where the action is centered around theft or violence. (That would probably cover a huge fraction of today's popular computer games.)
Maybe the world took a different direction after a September 11-style event, instead opting to further restrict not just aviation but also things like flight simulators.
These types of games or software could be restricted to a certain amount of play time per month or something like that. An easy way to do that is to only allow access to those games in government-run (or government-approved) arcade halls where play time is logged, centrally checked and logs regularly audited.
Restricted access would be harder to police in a world where fast, easy, worldwide data communications is available basically to everyone, but you could make the penalties for getting caught with contraband games suitably harsh and the probability of getting caught suitably high, which would dissuade most people.
Maybe game vendors simply never added multiplayer capability, or didn't expand on multiplayer capability beyond the most basic forms, to games available to ordinary people.
There is any number of ways you could plausibly make this happen.
Add a significant element (social, perhaps) to the arcade hall experience that cannot readily be replicated elsewhere.
Some ideas might be to offer unique multiplayer game experiences, or hardware that is so top-of-the-line and expensive that most people can't afford it, or some way and reason for people getting together just for the social activity and the game being merely an aside to that.
A dedicated arcade game hall that charges per game or per hour could, in principle, invest in hardware that would be beyond the reach of all but the most devout players. This, in turn, could allow players to play games that are far beyond the ability of most peoples' computers to run at an acceptable performance.
Consider that people still go to movie theaters, even though large-screen televisions, surround audio systems and high-quality digital copies of movies are available. Then, try to replicate the reasons why people go to movie theaters, but in the context of computer games instead.
TL;DR: make it so that arcade halls can (and do) offer players something that they can't readily get at home, at a reasonably affordable price, and demand is likely to remain.