Ah, no, a non-believer, hunt him/her, ahhhh....
Jokes. Depends what you want from a TV show, I guess.
Here's my tuppence.
I find it great escapism. Certainly daft (often unbelievably so) but in a good way. I watch it for the humour more than anything else, which luckily Moffat dishes out in spades. It has a great template for experimental story-telling, as well, so it's unique in that sense.
It's frustratingly unreliable quality-wise, too. If I lose interest any other TV show, I simply stop watching, but Doctor Who changes almost every series, every episode.
But I understand why many find it cringeworthy. Rory almost dying again at the end of the recent Pirate episode was one of the most hammiest, predictable and uninteresting scenes in Doctor Who I've ever seen.
Wish I could pick out a handful of absolutely perfect episodes to show you what it's capable of ... but one of the greatest (and in this case annoying) things about the show is that perfection is relative.
To answer your question, in short: Doctor Who isn't for everybody, but it's there for anybody.