I'm sure many people dislike Dr. Who for a variety of their own reasons (and they're certainly welcome to do so), but I think the show has some really great qualities if you give it a chance. I watched reruns of old episodes when I was a kid and I enjoyed them then (I felt at the time, on the strength of his wearing celery on his jacket, that Peter Davison and I should be friends), and I watch the newer episodes as an "adult" and enjoy them now.
I agree with many of the previous posts that you have to understand what Dr. Who does and doesn't aim to be. Dr. Who is silly and quirky and often ridiculous, and that's much of its charm. It's not plausible or reasonable, but that's not the point. Though it may sound trite, the heart of Dr. Who is really The Doctor. There are plenty of things that annoy me to no end about the show, plenty of times when characters or plot lines disappoint me, but I keep watching because he's such a compelling, contradictory character. He's childlike and effervescent despite being ridiculously old. He's excited about new possibilities despite having seen just about everything. He's interested in the universe despite being (intellectually and otherwise) totally alone. He's playful and hopeful even though he's been disappointed by the universe time and time again. He should be a tragic character, and in many ways he is, but I think because he's so energetic and surprising and silly most of the time his moments of real sorrow are (beneath all the silliness) particularly powerful.
I'm not a huge science fiction fan, but I think Dr. Who pulls off science fiction particularly well. For any story to be good, it has to both be surprising and true. Stories that are too mundane, that are too similar to our own experiences of the world, don't teach us anything. Stories that don't ring true don't matter to us because we can't see ourselves in them. Good stories, as anthropologists like to say, "make the familiar strange, and the strange familiar". Science fiction is naturally alien, so the challenge for good science fiction is to make the alien human. The Doctor does this really, really well. Despite being an alien with two hearts who fights monsters that are so clearly products of 60s it's both adorable and a little painful, he's a deeply human alien (not just because he's played by a human actor). The 10th Doctor once described himself as "laughing at the darkness", which I found to be a really beautiful and sad and fundamentally human thing to say. His character is ridiculous, but he's also uniquely poised to see other people's ridiculousness (which they tend to take very seriously), and challenges you to do the same.