Should I Show Them The Classic Series: You'd think that because Doctor Who started out as a kid's show it would be relatively easy to get kids interested the older stuff, but I've found it really difficult. At least in the three to five year range. Part of that is because we try to watch them all at once rather than spaced out like they were originally broadcast. That makes the average story over 90 minutes, which is a lot to ask of a kid.
Another mistake is to start at the beginning of any given serial. Most Doctor Who serials used a pretty formulaic approach where the first episode was all about setting up the location, and not bringing in the monster until the cliffhanger ending. You need to look at it more like Scooby Doo, and get the fright in right away. If you want to give kids a try on the classic stuff I recommend the second episode of any Hartnell Dalek story, or the final episode of "The Moonbase" on the Lost in Time DVD. "The Android Invasion" from Tom Baker's run is also good, and is a nice set up if you want to show them the definitely kid-friendly Sarah Jane Adventures.
Is the Show Too Scary: Sometimes it is and sometimes it isn't. It all really depends on the episode and your particular child. My daughter's been watching Nightmare Before Christmas since she turned three, as well as The Last Unicorn and that somewhat dark Spike Jonze version of Where the Wild Things Are. She doesn't scare easy.
Honestly, if I may return to the Scooby Doo episodes I mentioned above, a lot of classic cartoons can be pretty damned dark at times. Her favorite episode has a character whose soul is sucked out and is turned into a zombie. Kids tend to handle it better than we think, though of course if at any point they say they're scared you should of course turn it off.
What If They Don't Want to Watch It: Introducing kids to new entertainment is usually a chore. They like to watch the same things over and over again. Consistency is how they exercise control in their lives. Which is why you've seen that same episode of Team Umizoomi four dozen times.
In the beginning, don't make a big production out of showing episodes. Try to just have one on when they happen to be home rather than take the whole "You're going to love this" route because no matter what a kid is going to want to watch what they want to watch over what you want to watch 124 percent of the time.
Also, Dora has the right idea. Interaction is a great way to engage young viewers. When we watch The Doctor in our house my daughter gets to wear a fez and hold her sonic screwdriver. Whenever The Doctor uses his she uses hers to help him. Little things like that are a good way to make the experience less passive.
What If They Hate It: They might. Mine dislikes Fraggle Rock, which is a little heartbreaking to me, so it just stands to reason that some kids are not going to get into Doctor Who. You're best bet is to just acknowledge that and move on.
This is, however, a great moment to start teaching sharing of the TV. If they don't like the show, then take turns with them where they get 30 minutes of what they want and you get 30 minutes of what you want. Sometimes when you do this you find they fall into the narrative anyway, but if not you establish that entertainment choices in the house will be shared.
