Sesame Street is actually a pretty easy show to defend. When someone looks askance at me for liking it as an adult, I always say why the show has value for adults, rather than why I still watch a kids show. Depending on the person I'm trying to convince, I mention one of these reasons:
1) The writers know kids learn more if they watch with a parent, so they have always written it to be entertaining for the parents, too. There are all sorts of parodies, jokes, and cultural references that are specifically aimed at the grown-up audience. There are clips I remember liking as a kid, but now I know the songs and tv shows and celebrities they're parodying, so those clips are funny on a whole different level now.
2) It's really interesting from the standpoint of teaching, how they get different concepts across to kids. The creative ways they come up with to use humor and music and exaggerated characters to teach things from subtraction to cooperation, and make learning fun in the process, is interesting to think about and always makes me appreciate it more.
3) It's also fun to watch for artistic techniques (like in animations, songs, and puppet skits) and to appreciate the production and performance skills that go into making the show. I explain the "behind the scenes" stuff--how physically and artistically challenging it must be for the puppeteers to crouch on the floor, on some other puppeteer's lap, and hold the puppet up over his head and make it look lifelike and sing in its voice. Lots of people don't even realize the muppets are hand puppets, much less how the live hand ones take two people, or how they all do their own voices, or how the puppeteers play so many characters. And lots of people think the guy in Big Bird must be really tall--they don't realize the beak is a puppet, and that Caroll can't see out of the bird and he has to use a little tv and walk around seeing what the viewer sees rather than what he would see, etc. etc.
Whenever anyone makes fun or is skeptical and I respond with "actually, it's really smart/interesting/cool because...", they always see my point. Plus, because almost everyone watched it when they were little, whenever you give an example they even know the characters you're talking about