I understand your point about calling it mens and womens, and frankly, I am fine with it.
But just because I get weary of political correctness, I feel it should be clarified that the reason it is acceptable to call it “basketball” (without the “men” modifier) and “women’s basketball” is because women, if they were good enough, would be welcome to play on the team that is currently made up of only men (at least, I am not aware of any rules preventing it). The reason a women’s basketball team exists is because they need a new category that allows them to compete at an even playing field. Men are not welcome to play on the women’s team, but I am pretty sure the opposite is not true. So, basketball is just that, basketball played at the highest level by the University of Utah. Women’s basketball is a modified version that is restricted to just women.
The same is true of the NBA, we don’t need to call it the Men’s NBA. It is just a collection of the best basketball players in the world, regardless of sex.