There is no way to answer this question objectively. We can be more or less objective in laying out what the goals of the movement were, but there is no way to objectively say if they were legitimate. For the purposes of this question, I assume that you are talking about the women’s movement that started in the 1960s.
Let us first, then, lay out what the goals of the movement were. Of course, the movement was not monolithic in its goals. Not every woman or group of women was aiming for each of these goals. The main goal of the women’s movement was women’s control over their own lives. Women in the movement did not want to be dependent upon men or subject to control by men. Instead, they wanted to be like men in that they wanted to control their own lives and have the same opportunities that men had. This goal was manifested in a variety of ways.
Some of these manifestations were rather different from one another. For example, some women wanted to be held to the same sexual morals as men. In other words, they wanted to do away with the double standard that said women had to be monogamous but men could be unfaithful to their wives. Some women wanted to be allowed to have relationships with whomever they wanted. This was aided by the availability of birth control. On the other hand, women’s goals also manifested themselves in the push to be admitted to the (then men-only) Ivy League schools. They manifested themselves in the desire for women to have legal rights equal to those of men.
There is no objective way to say which of these manifestations of the overall goal were legitimate. To many people, the idea of women controlling their own lives and having equal rights is obviously legitimate while, to others, it is unacceptably radical, particularly when it comes to such things as sexuality or, as we have recently seen, women serving in the military.