Okay I will admit that something has always, not bothered but confused me about feminism.
It is, to my knowledge, the only equality campaign that has statistically significant backlash to it within the culture it is a campaign for.
If I get 100 gay men at random and ask them if the support gay rights, sure like with anything I might get a very small handful of opposing voices but safe to say the extreme majority of them are going to support it.
If I get 100 black people at random and ask them if they support black people... the same thing.
But, based on statistics if I pick 100 women at random and ask them if they are Feminist... I will get a statistically significant number of women saying they are not. As with everything exact numbers are hard to find but most polls only about 60% of women self identify as "Feminist."
And that's odd on purely cultural level. It's weird. And unlike a lot of other things they aren't really clear breakdowns on age, education, even political leanings. Sure only 7 percent of women who identify as Republican "strongly identify as feminist" but only 28 of women who identify as Democrat "strongly identify as feminist" (although the "mostly identify as feminist" spread is A LOT wider here, so that to me anyway says that agree or disagree, "feminist" is far less of a dirty word on the Left.)
Again there's no moral judgement to be had here, it's just weird is all.
So there is something; branding, emotional baggage, the way the wind is blowing, the price of tea in China, something that has tainted the idea (or the term, I wouldn't be surprised if this was one of those "The majority are in favor of helping the poor, but a minority are for welfare" kind of things) of feminism.
There are a couple of things I know about.
If we go back to the days of the Equal Rights Amendment, we see Phyllis Schlaffly, who famously rose up and opposed the ERA, and was the most prominent female opponent of the ERA. So, taking her for a shorthand version of anti-feminist women, I'll try to answer the question.
At the time, her biggest objection was that the amendment required that men and women be treated equally under the law. What that would have meant back in the 1970s would have been that a lot of laws which protected women would have been invalidated. In particular, in divorce cases, women were preferred in issues including child custody, property sharing, and alimony. The ERA would have invalidated laws or policies that favored women in those cases.
She wanted to ensure that if a man walked out on a woman, the woman did not lose her children, and the man still had to support, not just the kids, but also his ex wife.
A second aspect relates to shutit's themes in this thread, although it's not identical. I think one can reasonably say that feminism was an asepct of liberalism, and in this case I mean the things that would have been called "liberal" in the 1960s. A lot of people opposed that agenda, and they saw feminism as being strongly connected to that agenda.
This was especially true of the abortion issue. A lot of people opposed abortion. Feminist organizations pretty much universally supported abortion.
Similarly with gay rights. And then of course there were those bogeymen that people made up about the Equal Rights Amendment. It was all phony of course, but a couple of years ago I went back and read some anti-ERA literature from back in those days. Do you know that some people were saying that if the ERA passed, courts would end up saying that two men could marry each other, or that men could use women's bathrooms? Yeah. Ridiculous, right? How could anyone believe that kind of nonsense?
So, some women were opposed to feminism because they were opposed to the things that prominent feminists were aligned with.
I think, today, very few women, even very conservative women, would be opposed to laws forbidding women to adopt certain professions. I'm not sure what other laws that are considered "feminist" today that might be opposed by conservatives, especially conservative woment.