Well, I disagree. I'd prefer Doctor #12 to stay male. Let me explain before I have to go surrender my feminist card.
First of all, it's not what Doctor Who is. The story is about a lonely old man who picks up younger women (and the occasional younger man, which doesn't make it any less creepy), and they go off into time and space on an amazing adventure. He is a MALE character. That's who he is. If you changed the Doctor's gender, the dynamic of team TARDIS wouldn't be the same. Unfortunately, we still read certain gendered situations certain ways. We don't yet possess an understanding of gender that's nuanced enough to be able to swap them around easily. And so, there could be something analogous the the original premise, but it wouldn't be exactly the same. In a post-gender world, maybe, but today it would be a different show.
Second, why would the Doctor suddenly become a woman if he's been a man eleven times now? Neil Gaiman, in the backstory he provided for the Corsair, has established that Time Lord regeneration can change genders, but what people are ignoring is that he/she could identify with and thus become different genders, whereas the Doctor we've seen strongly identifies himself as male. There really is no reason for the Doctor to become a woman. As the Doctor changed from Christopher Eccleston to David Tennant to Matt Smith the character developed, so for the next Doctor to be a woman there needs to be a good reason in the story for this to occur. Now, I have no doubt that a competent writer could create a plausible scenario in which in which the Doctor would have to change gender, but I am far too skeptical that such a change would be a gimmick, especially in light of how patchy the last half-series has been. A female Doctor is certainly something people would tune in to see. I also suspect that a female Doctor would be used to promote how clever, how revolutionary, how forward thinking and willing to listen to the fans (but only the ones shouting on Twitter) the production team is.
And this brings me to my third and most important point. Seriously, read this even if the stuff above has pissed you off.
There should not be a woman Doctor until there is a woman head writer on Doctor Who.
Now, what I'm not saying is that men can't write female characters. Far from it. Moffat's women may come in four varieties (blonde, brunette, ginger, lizard), but plenty of men have written convincing, well-rounded women for Doctor Who.
No, a female Doctor merely disguises the fact that the production of Doctor Who has been predominantly male, more so since the show was revived in 2005. Since then, there have been four female executive producers of the show, but only two women have directed and one woman has written episodes. Furthermore, there hasn't been a single female writer on the show while Moffat has been in charge. And though the programme may have been launched by producer Verity Lambert, a woman has never served as head writer.
Until more women are given more senior roles in the creation of Doctor Who, a female Doctor would allow the production to call itself progressive while allowing such systemic inequality to continue. That is why we are not ready for a woman to play the Doctor.
One day, maybe the Doctor will be a woman. Or non-white. (Anyone else find it odd that there hasn't been an equally vocal bunch calling for a Black Doctor?) But not at the expense of story. And not at the expense of those who need to be given more opportunities on Doctor Who.