My recollection of the prime-time television I watched growing up -- the 70s and 80s plus whatever were on at the time -- are almost invariably of episodic television. At the end of each episode, everything that had happened was completely and forgotten, and the show effectively "reset" for the next episode. This included not just the obvious sit-coms, but also dramatic shows like sci-fi, cop shows, etc. I don't remember any show that stands out as paying the least attention to "continuity".
The only deviation here were soap operas, which were just a bunch of overlapping, long-running storylines woven together, but those seemed to be the exception to the rule, and only a handful showed up on prime time.
By the late 90s it seemed like there was a change, to the point where everything but sit-coms were more serialized. Episodes were still self-contained but also fit together into longer, multi-episode, season-wide or even multi-season story arcs. Things that happened in one episode would be written in to subsequent episodes on a regular basis.
For some reason, the two shows that always stick in my brain when I think about this are Golden Girls (one of the last shows I remember from living at home) vs. Buffy the Vampire Slayer (one of the first shows I remember from living on my own.) and how dramatically different those shows treated continuity. These days, most dramatic television seems to make at least some effort to be internally consistent from show to show.
Am I just mis-remembering the kind of TV that was popular up through the 90s? Were there a lot of quality serial-type shows on at that time that I just never saw or have since forgotten? Or was there really a shift in the type of programming that made it to prime time teleivsion, and if so, why did that happen?
(Also, I should point out that I'm specifically not counting things like major cast changes, which would be impossible for a show to ignore, but rather that events that happened in one episode are never brought up again, even when they would be relevant, or that there was no cross-episode plots that linked them together.)
Several comments have pointed out how much riskier serial dramas are than episodic ones from the network's perspective. That makes sense, especially when you factor in syndication deals later on. That might explain the my lack of memory of such dramas on television in previous decades. However, it doesn't explain why so many dramas on television are serialized.