adorksgeekyart answered: That is question that in nearly unquestionable. What is a Doctor without his Tardis? And a Tardis without her Doctor?
heartoftardis answered: They are equally important. You can't have one without the other.
b-lynninja answered: "We don't work that way" comes to mind.
eflauta answered: The Doctor.
strandedonamountain answered: The writing. Without the DOctor, we'd be watching The TARDIS
thetruthandthedream answered: The Doctor
The question was "Which do you think is more important to Doctor Who: The Doctor or the TARDIS?"
Now allow me to explain the question.
I was thinking last night about the importance of character to any story. Many people would tell you that without characters, a story simply doesn't work. As a Doctor Who fan, I began to think about the importance of the TARDIS to the Doctor, as a character and in the grand scheme of the story that is Doctor Who. I know, of course, that the TARDIS is herself a character, as was confirmed in the brilliant episode The Doctor's Wife. However, for the purposes of this, I shall be looking at the TARDIS as part of the plot, a plot device.
Going back to the very beginning of Doctor Who: An Unearthly Child opens with the distant sounds of a bell, a police officer is walking through the foggy night, opening the gates to a junk yard to ensure that everything is as it should be. He shines his torch around, and sees a police box, standing in the junk yard. (Indeed, in the Target novelization of An Unearthly Child, the police officer wonders why it's there, but then concludes that it must be old and no longer functional.) All Doctor Who fans are well aware that this police box is actually a TARDIS, a magnificent time ship that can take one anywhere and anywhen. This time ship, this TARDIS, belongs to a mysterious character known as the Doctor, and only as the Doctor.
Doctor Who literally begins with the TARDIS. Doctor Who is the story of the Doctor, true. Sure, the story of the Doctor as a whole begins, of course, with the Doctor, but our view into the Doctor's Universe begins with the TARDIS.
Which leads me to this: would the Doctor be an interesting character without the TARDIS?
I would say that the biggest appeal to the Doctor is not his advanced, alien intelligence, or his eccentric personality. His greatest appeal is the possibility that being around the Doctor brings: the chance to do everything, see anything, go anywhere. And that would not be possible without the TARDIS.
However, think of the Third Doctor's era. The majority of this era was spent on Earth, as the Doctor could no longer pilot his TARDIS. Did the show continue? Of course it did. The show even enjoyed a surge of popularity, bringing in viewers who weren't as fond of the alien ships and planets of Doctor Who until that point, but thoroughly enjoyed the drama playing out on the familiar planet Earth. All of this, was without the TARDIS bringing the Doctor to a new place every week.
The reason I asked the question, "Which do you think is more important to Doctor Who: The Doctor, or the TARDIS?" was to see what other Whovians thought about this. My own opinion is that they are equally important. As adorksgeekyart said, you cannot have the TARDIS without the Doctor, and you cannot have the Doctor without the TARDIS. They are both equally important, and the show would be nothing without either. Did the Doctor spend time without access to the Universe via the TARDIS? Yup. But it was his experience of the Universe that aided so much with the alien invasions that the Third Doctor's era (The UNIT era) featured. I'd say that the majority of this experienced didn't come from his Time Lord education, but from his experiences travelling the Universe.
Again, thank you for the responses, and if you've read this to the end, thank you for dealing with my rambles.