Well, if we want to be objective about this I'd probably have to say "Mobile Suit Gundam", or "Neon Genesis Evangelion".
The reason I say this is because the original Gundam has endured such a ridiculous length of time and arguably inspired everything that came after it to some extent, as well as moving Mecha into popular culture. Albiet that popularity is mostly present in Japan, despite a lesser mirroring internationally.
"Neon Genesis Evangelion" is noteworthy because of the fact that it's responsible for catapulting anime into the international community in a big way. A lot of periodicals you read will even divide the anime industry as being pre-and-post EVA given the massive way it changed things both in terms of production, ideas, and of course marketing and distribution. It's also noteworthy because by all accouts EVA was a complete mess, where apparently the team doing it was told they were going to get more than one season, and then part way through production were told that this was not the case because the guys with their hands on the purse strings didn't think it was going to go anywhere. When it did, the giant temper tantrum that was the ending, along with all the dropped plot threads, became a very big deal, and actually did a lot to increase the fame of the anime.
I'd probably give honorable mention to one of my favorite franchises "The Dirty Pair". Despite not being one of the more prolific equasions, this series pretty much launched the whole female anime protaganist into what it is now. The entire "Girls with Guns" genere including things like "Noir" ultimatly owes it's existance to this anime. Not to mention that international syndication (albiet not to the US) introduced Anime to a lot of the world, even before they knew what it was exactly. I suppose you could mention "Sailor Moon" alongside this (which I'm not a huge fan of), but I think ultimatly it was less influential, it launched the "Magical Girls Genere", but I think without what "The Dirty Pair" established for female heroes/super heroes beforehand, you never would have see that kind of take on things.
Things like "Cowboy Beebop" are getting old enough now where I suppose they could be considered classic, but I don't think they had that large an influance on Anime as a whole. From right around the same time period (when VHS was beginning to fade out, so you saw Anime in both formats) you also had things like "Vision Of Escaflowne" (the TV series), while not a huge success it's noteworthy that Escaflowne, while destroyed due to editing, was a big enough deal where that was the show Networks were picking up for a US release on TV (Fox) it wasn't until later on down the pipe you saw "Beebop" getting much exposure outside of the hands of fanboys (like me) who were at the time willing to dish out $30 for a couple of episodes, or knew some good fansubbers.
This is all simply my opinion though.
