If that's really true, then the Doctor should be ashamed; true intellectuals ought to enjoy mental stimulation from others, not feel threatened by it. But it's not true, because the Doctor demonstrates time and again that he admires his companions for their intelligence, and not their fashion sense and stylish hairdos. The Doctor travels with smart women and tough women, and they contribute quite a bit to his heroic efforts.
I've already said plenty about Barbara's character in my episode reviews, but to recap briefly, she is clearly one of the most valuable characters in the entire series. She saves the entire TARDIS crew in The Edge of Destruction, she becomes a goddess in The Aztecs, and she demonstrates her resourcefulness by fighting El Akir without the Doctor’s help in The Crusade.
After Barbara, I must confess that there is a string of fairly useless female companions. In The War Machines, Dodo and Polly are little more than WOTAN’s pawns; for dramatic purposes, they exist solely to get hypnotized and lure the Doctor into danger.
There are a ton of silly myths circulating about Doctor Who, and one of the most common – and least accurate – is the notion that all of the show’s female companions are stupid and useless. This idea pops up in More than 30 Years in the TARDIS, where the Doctor's companions are likened to "James Bond’s bikinied lovelies, no more, no less." Steve Lyons added fuel to the fire in a Doctor Who Magazine article a few years back, when he suggested that the Doctor actually enjoys the company of empty-headed women because they make him look cleverer by comparison.
Granted, Victoria will never be mistaken for a woman Rambo, but she represents an important step towards building up the respectability of the female companion. Zoe, her successor, demonstrates in The Krotons that her IQ is on a par with the Doctor's. In The Invasion, Zoe gets to put her mathematical intelligence to use as she calculates how to destroy the entire Cyberman fleet with a handful of missiles. Surely blowing up a huge invasion force qualifies Zoe as being "useful," doesn’t it?
Liz helps the Doctor construct the Auton jamming device in Spearhead from Space, and she aids him in curing the Silurian plague in the following episode. Jo Grant starts out a bit useless in Terror of the Autons, as she is hypnotized by the Master and used against the Doctor, but in the later episode Frontier in Space she demonstrates that she has grown stronger by resisting both the Master's hypnotic gaze and his hallucinatory machine. Sure, the Master outwits her in the end, but that's because he's a 700-year-old (or thereabouts) Time Lord...if Jo totally made a fool out of him, he’d be tough to respect afterwards. Sarah Jane is the most useful in episodes written by Robert Holmes, her creator; she leads a raid in The Time Warrior and blows up Sutekh's war missile in Pyramids of Mars.
Leela is quite possibly the show’s most valuable companion, predating similar characters like Xena and Buffy the Vampire Slayer by about twenty years. In Leela's introductory episode, The Face of Evil, she is cast out of her tribe for doubting their religious convictions, which is in itself a sign that she has an independent and intelligent mind. Later in the episode, Leela figures out that Xoanon’s voice must be emanating through a gap in the time barrier, impressing even the Doctor with her insight. "You’re a genius," he gushes, "a genius," and I could swear that he looks ready to kiss her. This is the guy who likes being around stupid women?
Leela is also useful in a purely physical capacity (wow...there's a sentence with an unintended double meaning! Sorry...). She saves the Doctor on numerous occasions, from savage tribesmen, Sontarans, and Tong assassins. Unfortunately, after Leela, the female companions start to lose their importance again. Romana is clever but inexperienced, and despite her intelligence she rarely does anything that important. Nyssa and Tegan's roles vary tremendously based upon the writer, but generally speaking, they don’t play too large a part. The same can be said of Peri and Mel, though in Planet of Fire Peri does outwit the Master for a while and give him a good fight for control of Kameleon. As a whole, the 1980s episodes are a time of backsliding for women in the show.
The TV Movie tries very hard to introduce an impressive companion in the person of Grace, but it tries perhaps a little too hard. In order to demonstrate her sophistication, Grace acts unimpressed when she sees the interior of the TARDIS because she understands the theory behind transdimensional engineering – a cute joke, but totally unrealistic in my opinion (she should be amazed anyway). For a supposedly tough woman, she looks pretty wimpy as she whines and clings to the Doctor while he zooms around on a police motorcycle in a conspicuous display of alien testosterone. She redeems herself to an extent when she saves the Doctor by short-circuiting the Eye of Harmony, but since her actions consist of sticking random wires together until the Eye malfunctions, she comes across more as being lucky than being smart.
New companions are introduced all the time in other mediums. Bernice, Charlie, and Sam are but a few of the women who accompany the Doctor in comic strips, books, and audios. On the whole, their roles are similar to their television counterparts; they are valuable companions to the Doctor but they never overshadow him, partially because it’s tough to overshadow a near-immortal alien genius, and partially because it is, ultimately, the Doctor’s show.
The ladies in the Doctor’s life, taken as a whole, have had a bumpy but mostly rewarding career. A few of them are useless. Some are smart, some are strong, and the lucky ones are smart and strong. In the final analysis, the vast majority of the Doctor’s female companions play important roles in the show, and virtually none of them are just window-dressing. The Doctor, I think, rather enjoys their intelligent company.