Why were some Americans not in favor of women's equality?

The simple answer to your question is that many people are afraid of change, especially when that change results in an upset of traditional power structures, in this case, patriarchy, or male dominance.
Americans are not alone in some showing resistance have against the advancements of women. Sexism and misogyny exist all over the world. Since your question is in the past tense, I will assume that you are talking about the advances women made during the first-wave feminist movement of the mid-nineteenth to early-twentieth centuries, when they gained the right to suffrage, as well as the second-wave movement of the late-1960s and 1970s when women made gains in terms of reproductive rights, education, and employment.
Because American society has historically been composed of certain groups being dominant while other groups are implicitly subordinate, it is difficult for many people to consider a society in which everyone has equal access to opportunity and resources. Therefore, when women demand "equal pay for equal work," many choose to hear that as women possibly taking away from their financial gains.
Hillary Clinton recently told a story about taking her law school entrance exam in the late 1960s with a few female friends and classmates. The young women sat in a room full of men who jeered at them. One even yelled out that if one of the young women took his place, he would be sent off to Vietnam. There was a sense of entitlement in his statement of "my place." 
Resistance to women's reproductive access is more morally complicated, frequently with religious authorities weighing in. However, here, too, there is a fear of women moving too far out of their traditional roles as wives and mothers. Women who are not limited to the home become competitors at school and in the workplace. They also become less sexually accessible to a single man, which upends traditional monogamy a bit.