One of the mothers, defending the infant she could not properly feed, gave him an angry growl in return; he lacked the energy even to cuff her for her presumption. (1.4)
The man-apes appear to be very patriarchal; the guys are in control. This may seem like an anthropological truth—folks often assume that earlier people, not to mention earlier apes, were male-dominated. In reality, though, many researchers believe that social relations among hunter-gatherer peoples were often very egalitarian—strict gender divisions in labor and power only came along later. Some of this research is since Clarke's time. But still, it's worth realizing that the vision of early man as male-dominated is not necessarily a truth; it's a story he's decided upon.
Part 1, Chapter 6
Ascent of Man (title of Chapter 6)
Throughout the book, the novel uses "man" to mean "human beings." This is most noticeable with the "man-apes," not all of whom are men. Using "man" instead of human is supposed to sound more impressive—but it also reflects the extent to which there just aren't women in the book. The explorers, the inventors, the part of humanity that ascends up the evolutionary ladder—all those folks we see, just about, are men. As far as the book is concerned, it is the ascent of man. Women are off taking care of the kids, and doing whatever else they do while men head for the stars.
Part 2, Chapter 7
"Thank you," said Floyd with a smile, wondering why stewardesses always had to sound like robot tour guides. (7.24)
Even in the future, the stewardesses are all women. Floyd wonders why she has to sound like a robot—but the answer is that she, like all the women in the book, doesn't get to be a real person, she's just window-dressing.
"Dr. Floyd," demanded a very short and determined lady of the press, "what possible justification can there be for this total blackout of news from the Moon?" (7.14)
The lady of the press is about the only women in the novel we see who does not have a job like "stewardess." The reporter here is presented slightly satirically ("very short and determined"), and as an annoyance, trying to ferret out gossip from the competent, tight-lipped Dr. Floyd. Men do important manly things in space; women bother them.
Part 2, Chapter 9
…she came from Bali, and had carried beyond the atmosphere some of the grace and mystery of that still largely unspoiled island. (9.43)
One of the stewardesses on the moon-flight is from Bali, and she does a dance for Floyd. Bali is presented as primitive, and the woman is linked to that primitiveness. Floyd is the manly dude flying into space, contrasted with the sensual, exotic woman doing her sexy dance for his pleasure and entertainment.
Part 2, Chapter 10
Floyd found himself back in the familiar environment of typewriters, office computer, girl assistants, wall charts, and ringing telephones. (10.21)
Again, "girl assistants" are part of the furniture, like "wall charts" or "ringing telephones". Clarke's future is mired in the gender assumptions of 1968. Other science fiction of the era (like Ursula K. Le Guin's Left Hand of Darkness) imagined gender reconstructed and rejiggered. Not Clarke though; for him, it's all still about the girl assistants.
Part 3, Chapter 17
It was true—indeed notorious—that seamen had compensations at other ports; unfortunately there were no tropical islands full of dusky maids beyond the orbit of Earth. (17.27)
This is the second reference to sexualized non-white women (after the dancing stewardess from Bali) in the book. The book is almost completely sexless, but it seems significant that the times when sex is almost/sort of mentioned, it's in the context of "exotic" women providing various services for space travelers.
Though numerous ladies had promised to wait until the expeditions returned, no one had really believed this. At first, both Poole and Bowman had been making rather intimate personal calls once a week, though the knowledge that many ears must be listening at the Earth end of the circuit tended to inhibit them. Yet already…the warmth and frequency of the conversations with their girls on Earth had begun to diminish. (17.26)
The women are not only irrelevant to space travel; they're also fickle. Or perhaps it's that, to be real adventurous space travelers out among the stars, you need to cut yourself off from the distractions of those ladies and girls bound to earth.
Like all his colleagues, Bowman was unmarried; it was not fair to send family men on a mission of such duration. (17.26)
Why couldn't they send married couples into space? And why is it all "family men"? The assumption seems to be that women wouldn't want to go, or wouldn't be fit to go. Again, the gender relations of what is supposedly 2001 seem mired in the '60s, or even the '50s. Clarke likes to imagine changes in technology and whooshing spaceships, but changes to gender norms seem much harder for him to manage.
Part 6, Chapter 45
In an empty room, floating amid the fires of a double star twenty thousand light years from Earth, a baby opened its eyes and began to cry. (45.11)
Science-fiction started with a birth-without-women—Dr. Frankenstein's creation of an artificial monster. The star baby is also a kind of new life made without recourse to sex, pregnancy, and women. Part of progress in the novel is leaving women behind; you don't need them anymore. In this novel, women in the novel are mostly associated with earth and bodies; men with traveling to the stars and escaping from bodies. You could see 2001 as an extended explanation that men are jealous of childbirth and want to make their own kids and are going to take their toys and go out to the stars until they can get pregnant too, darn it. We're not buying it, though.