In the current system, schools need to have an equivalent number of sports teams (and slots) for men and women, with the same number of scholarships (the law governing this is called Title IX). Generally speaking, women's sports don't bring in money for schools, whereas football at a competitive school will bring in money in the form of TV deals and even donations from alumni. So essentially, the football and basketball teams subsidize the rest of the athletic department. If you had to pay the athletes in these sports, then there'd be less money to go around for other teams that don't generate revenue. But at a minimum, you'd still need a few women's teams because of Title IX.
EDIT: to clarify, I personally think it's ridiculous that star college athletes don't get paid given how much time those athletes put in and how much money they bring in for schools, but I also think that at least at the schools with huge athletic programs it'll have some effect on other sports.
EDIT: to clarify, I personally think it's ridiculous that star college athletes don't get paid given how much time those athletes put in and how much money they bring in for schools, but I also think that at least at the schools with huge athletic programs it'll have some effect on other sports.