I'd say any time between the mid to late 90s for me, but to really narrow it down I'd say maybe 1997, after the success of First Contact DS9 & VOY really had some great runs. Notably the ends and beginnings of seasons in that year.
DS9 with the s5 finale Call To Arms and s6 opening with that great 6-parter, and Voyager with the awesome Scorpion 2-parter over s3 & 4, and the whole general reinvigoration the show had with Season 4 with the addition of Seven of Nine.
And I know special effects doesn't always equal good necessarily, but there were things in episodes around that time you started seeing that you'd never seen in TV Trek before. Whether it be all the DS9 battle scenes, Species 8472, great shots of Voyager landing on the planet in Demon, the crash in Timeless, etc etc
Of course I only started watching Trek in 1994, so I can't really comment on the years prior watching episodes "live." So I couldn't say on general popularity before that, but it felt to me around between and around Generations & First Contact in the mid 90s there were a lot of things going on- the big movie of Kirk meeting Picard, DS9 really picking up with the introduction of the Dominion and the Defiant, the launch of Voyager, Worf joining DS9 and the whole "the Klingons are bad again" got a lot of press attention, the 30th anniversary, First Contact getting great reviews and even non-Trek fans loving it, etc etc
That really was a great time to be a Trek fan