I'll throw in my two cents on this, because I've had a lot of issues with the writing since Moffat took over, the subconscious sexism only being a part of it.
A lot of the bad writing I think can be less attributed to overt sexism, and more attributed to Moffat hijacking the Doctor as his own personal wish-fulfillment alpha male avatar. Starting with "The Eleventh Hour", the focus of the show shifted heavily into the realm of "Look how awesome the Doctor is, everyone love and praise him!"
I mean, look at the introductory episodes for Rose, Martha, and Donna. Each of them opens with a lengthy segment following each woman on her daily routine and home life. The Doctor often doesn't show up until at least 10 minutes into these episodes. By that time, the companion as a person has already been established, independent of her relationship to the Doctor, and the remainder of the episode focuses on how her life changes upon meeting the Doctor. Upon meeting the Doctor, the companion demonstrates initiative by saving the Doctor without needing to be instructed, and her actions impress upon him and he asks her to come along.
Move on to "The Eleventh Hour". The Doctor is the first person we see in this episode, and we only ever see Amy when he interacts with her. Who was Amy before she met the Doctor? What did she like to do? We only get her characterization through quick bits of offhand exposition rather than actually experiencing these things for ourselves. Throughout the remainder of the episode, Amy completely lacks agency. All she does is follow the Doctor around. The day is saved because the Doctor tells everyone else what to do and they do it. They're all just agents of his will. In the end, the Doctor takes her along not because he's interested in her as a person, but because he's curious about the crack in her wall and the effects it's had on her.
Move on to "The Bells of St. John", and Clara fares a bit better. We at least see her interacting with other people and going about her daily business outside of the Doctor's influence, however briefly. She also manages to show initiative by hacking the wifi network and finding the evil hideout. However, the Doctor still primarily takes her along because she's a "mystery", not because of her as a person. Thankfully Neill Cross picked up a lot of Moffat's slack in "The Rings of Akhaten" and greatly fleshed out Clara as an independent character.
And then there's River who, again, doesn't hold the Doctor's interest because of who she is as a person, but because of the mystery surrounding her. Davies' Doctors obviously cared about their companions as people, but Moffat's Doctor just sees them as puzzles to be solved, and once he's got them figured out, then they're only good for following him around and praising him.
There's also the problem that Moffat's companions aren't allowed to have any jobs or interests that don't directly relate to being a woman or pleasing a man. Amy is a kissogram, a model, a wife, a uterus to create River Song. Clara is a barmaid, a governess, a nanny. Clara at least has a (stated) interest in travel.
Also notice how the Davies vs. Moffat companions dress. Rose was often sporting fancy make-up and hair, but her clothing was usually downright frumpy. Her first two episodes had her in sweatpants and a hoodie, and after that she was almost always in jeans or slacks. Martha and Donna were the same. It wasn't until Amy and Clara that the companions decided that heels and a mini-skirt were optimal dress for traveling the universe.
Then there's the Moffat-era women's ubiquitous obsession with the Doctor. As I mentioned, this is partially sexism, but mostly Moffat just using the Doctor as his own wish-fulfillment avatar. His Doctor is the guy who struts around everywhere, beating all his enemies on reputation alone by giving big speeches about how badass he is without having to do anything to actually prove it, who has women falling over him left and right, who has men grumbling and shuffling their feet around him because they can't be that awesome, and who never has to learn from his mistakes.
And that's what I think really bothers me about Moffat's themes instead of Davies. The main theme of Davies' era was how the Doctor influenced the people around him to be stronger and better themselves. When the characters give long monologues, it's often about how the Doctor has shown them how much more there is to life, such as Rose in "The Parting of the Ways" or Donna in "Partners in Crime" or even Elton in "Love and Monsters".
Contrast to Moffat, and the side characters all seem to be in a race for who can be the most obsessed and devoted to the Doctor, at the expense of everything else. Played up to the extreme when River was willing to sacrifice the universe for the Doctor's sake in "The Wedding of River Song". Female characters especially love to give long, doting monologues on how amazing the Doctor is. This idea of the Doctor tempting people to throw away everything for his sake was played out egregiously at the end of "The Angels Take Manhattan" when the Doctor takes it as a given that Amy will abandon Rory to his fate and come with him instead, and has a complete breakdown when she refuses (it's telling that he warns her "If you go, I will never see you again", implying that he believes he is the most important thing to her).
This idea that the Doctor is the most importentest thing in the entire universe is seen in all three of Moffat's season finales. Unlike Davies' finales, we never really see what effect the conflict is having on anything outside of the immediate main cast. Davies had Daleks wiping out all the humans on a space station, the Cybermen breaking into homes, the Master wiping out 1/10th of the world's population and taking it over, the Daleks rounding up people and experimenting on them. In "The Big Bang", the Doctor just offhandedly mentions that the rest of the universe and humanity is gone and they're all that's left, but you wouldn't have known that if he hadn't told you because no issue is made of it. The primary conflict is "Oh noes, the Doctor can save the universe, but he'll be wiped from existence! The Doctor or the universe, that's a hard choice!" In "The Wedding of River Song", the universe gets thrown out of whack, but nobody seems particularly bothered by it other than, "Huh, this isn't right." But the primary conflict is, "Oh noes, the Doctor can save the universe, but he'll die! The Doctor or the universe, that's a hard choice!" And then in "The Name of the Doctor", this trope almost goes meta when the main focus again is on the safety of the Doctor, and Vastra belatedly runs outside just to exclaim, "Oh yeah, and the rest of the universe is ending, too. But that's not as important as OMG THE DOCTOR!"
To end this excessively long post, I'll just sum up with: While it's true that sexist tropes are more prevalent in Moffat's era over Davies', the female characters' excessive submissiveness, obedience, and clinginess can be more readily explained by the Doctor's promotion to masculine godhood.
