A lot of you are mentioning Sonic. I don't think Sonic counts as a series that went downhill on account of it was never any good in the first place. To go downhill requires at least being somewhere nearer the top of the hill at some point. And if you disagree and say "Well the old Sonics were good", I urge you to go back and replay them. Seriously. They weren't good.
There were a few good aspects of some of the games, such as the music, or certain boss battles, but overwhelmingly the gameplay was long boring stretches with little actual control. Pay attention to where he's looping around for example, which constitutes majority of those games. Are you controlling any of that? Not really. Boring.
And if we're going with storyline aspect, original story is something like a mad scientist captures animals to have them work machines or something, so a hedgehog jumps on them to free them. Is that a good story? No it's an incredibly boring story full of obvious plotholes such as why the hell would anyone, even a mad scientist, choose to do this (what's in it for him?) and why his robots are so poorly made that even a small animal jumping on them breaks them.
So it was never good in gameplay. It was never good story-wise. The only thing that might be considered good is some of the music.
Now, game series that actually qualifies? Someone mentioned Mega Man and I would agree with that. IMO, Mega Man "jumped the shark" as soon as they started kiddifying everything. Compare Mega Man from the box art of Mega Man 2 with, say, the kiddish Mega Man in Mega Man 8. Give us some master robots that like blowing stuff up, not a bunch of "cutesy" kid stuff.
Some are saying Mario went downhill. I disagree there. I think Mario has done quite well over the years and is one of few that has managed to stay relatively stable. Sure there's questions of just how many kids does Bowser have, what was the origin of the Mario Bros. themselves, and related inconsistencies, but compared to other series he's done quite well. He did a hell of a lot better than Mega Man who somehow went from adult "50's futuristic" robot to some annoying kid robot. I mean there was the whole baby Mario in Yoshi's Story but that was supposed to take place in the past and he's not still a baby Mario. That's more of a one-off although I realize there are a few other games featuring baby Mario. In the overwhelming majority of games, he's good ole regular Mario.
Really why are things babied down or kiddified anyway? It can be an otherwise great game but deep down some part of me is going to find it annoying if they do that and if I enjoy the game in spite of this it's only because everything else about it is a great game, but it's a big ding. The main hero or character doesn't need to be a "kid" and especially not a baby. Stop doing that!
I mean holy crap when I was a kid, we didn't like pretending we were kids and especially not babies because, yaknow, we were already kids and why in the hell would we want to go back to being babies?! So it was much more fun if the superheroes were adults. Even if it wasn't realistic, you could pretend like maybe you could be like that someday and you could look up to them. Shows like G.I. Joe and He-Man. Nowadays it's like every other thing is kid heroes this and kid heroes that. Where's the fun in that? Only a lame kid would rather play pretend they're a kid hero than an adult one.