It wasn’t the internet, but on an episode of ALF, Alf used Willy’s computer to do online stock-market trading. According to wikipedia this was in season 4, episode 4, in 1989.
I’d call this a borderline case. Given that the characters are college students, they could be dialing into the internet with a phone modem, or it could be a direct peer-to-peer connection over the modems. From the dialog given, it isn’t clear that it’s the internet.
I haven’t seen that episode, but it could be a winner. Much earlier than my nomination, “I, Robot… You, Jane” from season 1 of ‘Buffy the Vampire Slayer.’ April 28, 1997
The episode, “Divine Recall,” of Picket Fences (Aired February 11, 1994) featured a naughty movie of the Mayor uploaded by a kid to the “World Wide Web.”
Diagnosis Murder had an episode Rear Windows '98 that had Amanda witness a murder happening on a webcam. May not be the first internet reference but perhaps the first to use a webcam as a plot device.
My ignorance is being fought here as to what does and doesn’t constitute the internet. I think the Picket Fences example is the earliest unambiguous one so far.
I think the Simpsons reference listed above wins. As someone who was using it in College myself in 1993, what they are describing was the internet. It was basically Gopher databases and Usenet newsgroups.
To be fair, though, IIRC, they used the term “World Wide Web” interchangeably with Usenet groups and BBS terminology. I suspect the writers had heard about Al Gore and Bill Clinton talking about the Information Superhighway the previous fall, and didn’t fully understand what they were talking about.
Another definition question.
In Fail-safe and Dr. Strangelove, one of the technologies on display was a computer system displaying (on the “big board”) radar inputs and radio communications from multiple geographically distant sources.
This is a “network” of (defense) computers. But since the regular public doesn’t have access to this, it’s disqualified from the OP, correct?
Does Max Headroom count as an Internet-based TV show? What about Automan?
Was Max Headroom about the internet, or about a totally computer generated image/character?
The little kid did a lot of hacking into power grids, control circuits, surveillance systems, etc., all from the confines of his laboratory, as I recall.
By direct connect modem?
I’m not picking on the show. I never saw it. 
Obviously not the first, but just to add a data point:
In the Millenium episode “The Mikado,” a killer streams video from his lair to the internet. When the page’s view count hits the target number he’s written on the wall behind his victim, he slits her throat on live video. It first aired February 6th, 1998.
(I remember this episode clearly, both because it’s the only time I watched the show, and because the same plot was ripped off for a movie a few years ago.)
Damned if I know! I don’t think they ever explained it; he just could do it.
But is that really a plot device?
I would think they are disqualified because they both pre-date the internet, which started on October 29, 1969 (as ARPAnet).
On Oct 3rd, 1995 the show Nowhere Man starring Bruce Greenwood was an episode called “A Rough Whisper of Insanity.” Get it? It’s an anagram of “information superhighway.” In the episode Greenwood’s character meets a “reclusive computer genius” who through the mysteries of the worldwide web manages to find clues about Greenwood’s secret past. I remember his last words before the bad guys killed him: “IT’S NOT WHAT YOU THINK!!”
Nitpick: “A Rough Whimper of Insanity.”
Good one.
We need to pinpoint when the term ‘internet’ was first used, then when ‘The Internet’ was first established in order to give an earliest boundary value for the search.
The word ‘internet’ was first used as an adjective in the Network Working Group: Request for Comments: 675 (RFC675) titled ‘SPECIFICATION OF INTERNET TRANSMISSION CONTROL PROGRAM’. This was where TCP first was defined. The date is December 1974. The word was shorthand for ‘internetworking’.
It took the establishment and standardization of TCP/IP before the term ‘The Internet’ was first used. This happened in 1982. NSFNET came online in 1986. This was the first time that anybody but a few elite CS guys in a few scattered American universities could actually use the Internet.
So, the earliest possible year for anyone using the term ‘The Internet’ is probably 1986.
I know some of this from online research and some from personal experience. However, movies are not my bailiwick. I can only hope that this information may allow a more precise search.
The Season 2 finale of Friends, airing May 16, 1996, featured Chandler on an internet dating website. Might not be that early, but it did have significance to the plot.
Wow, I was only thinking of this show last night. I couldn’t recall the name of it.