Doctor Who has been around for almost my mother's entire life--and I'm an adult. That should tell you something. Since 1963 the values and appropriateness of the show has varied alot. I'll tell you that the earlier stuff is more appropriate than now for most part, and keep my commentary mostly to the new series.
Doctor Who started as a family program and it still tries to be that. The story of an ancient, lonely alien who travels through time and space helping people who need it. He has trouble without someone to bounce his ideas off, and he's lonely, so he almost always has a companion, almost always a brilliant or brave young woman.
If you have a teenager you needn't worry about Doctor Who. There are the odd racy comment, and a fair amount of violence, but it's nothing your teenager hasn't heard on the school bus, and is often much more elegantly phrased and genuinely funny on the show anyway.
My concerns for younger, tween viewers are a few. The biggest iis its mild sexual comments. It's almost all a handful of jokes here and there, but they are subtle jokes that kids might accidentally repeat without a little clarification--and do you really want to explain the other definition of those words? They still aren't that bad, but let me tell you, being the naive kid to repeat that stuff without knowing it is not fun.
The violence can be fairly intense, but is usually contained enough. Remember, this is prime time programming in the U.K. The Doctor generally avoids all but the most temporary, emergency violence, preferring to talk, scheme, or sabotage his way out of trouble. His companions do not alwayts agree and there are traditional military battles in later seasons. Creatures are occasionally killed, sometimes violently and callously, and the series features at least one suicide, on-scene but off-screen.
Also, expect a lot of mild swearing. Nothing too crazy, but if you don't want your kid spouting the midlest but most religious of four letter words, talk to them before watching.
On the bright side, the show features great role models like the Doctor (courageous and resourceful) and his companions (especially Martha, a brilliant and brave woman who knows when to follow along and when to stand up, and Rory, a loyal, courageous friend unafraid to speak up for his beliefs). The Doctor is a flawed role model, and his character arcs in season 2 and 5 are too complicated for younger teens without a parental explanation of deconstruction or a close watch of "The Waters of Mars" and "The End of Time." The show advocates standing up for what's right and heping others, but is also careful to remind us to be careful in our meddling and to pick our batles wisely, with the way fixed points in history cannot be changd. It is a deep, thought provoking show best saved for older teens, but fine for viewing by younger ones and a good conversation starter for tweens and parents.
Plus, David Tennant rambling at fifty miles an hour and Matt Smith insisting that bowties are cool. Fun for the whole funny family!