The Common Sense Media review of the new series (as recommended in this question) suggests:
Not age appropriate for kids under 8, age appropriate for kids over 10; suggested age 10.
There are really five sets of Doctor Who TV:
- K-9 - this is aimed at children, and generally plays on channels aimed at children (e.g. Cartoon Network or Disney Channel). The show isn't very scary at all, although it's probably not very interesting to children under about 6, or anyone over about 16.
- The Sarah-Jane Adventures - much like K-9 (i.e. not scary), but for slightly older children (perhaps 8+), and more watchable by adults. Note that since Elisabeth Sladen (who played the titular character) passed away this year, there will presumably be no new episodes of this show.
- Doctor Who (original series, i.e. 1st through 7th Doctors) - this isn't a children's show, it's a family one. The difference is particularly in "scariness" and appeal - children and family shows will both avoid violence, sexual material, and so on, but a family show will have more complex story lines and may be too scary for young children, especially alone. The majority of the scary parts in the original series came from the villains (e.g. the Daleks), which are (of course) rather dated-looking now. If something that looks like a monster isn't scary, then the original series is probably fine.
- Doctor Who (new series, i.e. 9th Doctor onwards) - like the original series, a family rather than children's show. Although some of the villains here are scary-looking, there's also more psychological terror than the original series had. The result is that it's often less scary (than the original series) for young children (e.g. under 5), but more scary for older children (5-8), because they are able to understand more of what there is to be scared of.
- Torchwood - aimed at adults only. There's plenty of sexual content and violence, although I wouldn't say that it's particularly scary.
The new series varies a lot in the tone. Some episodes are very light and there's little, if anything, that would scare even a child. Others, like Blink, could scare an adult.
I've watched many episodes of the new series with my four-year-old. However, I've only watched repeats with him, and there have been a few episodes that I've skipped past. This is a good way to enjoy watching the show together, without having to worry that a terrifying moment might be around the corner (and since you know your own children best, you're in the position to make that judgement).
If it turns out that this doesn't leave many episodes in your children's case, or if you'd prefer to watch something you haven't seen, then watching The Sarah-Jane Chronicles (or K-9 at a stretch) together is a good compromise - you get to enjoy the occasional references and tie-ins with Doctor Who itself, and when they're a bit older they'll be able to move on to Doctor Who and learn more about the stranger that Sarah-Jane is always talking about.