The women do race the same courses as the men at some venues, Lake Louise and I think Val D'Isere being two of them, while the recent women's Aspen DH course retained quite a bit of the original men's course. But those courses are relatively easy by men's standards compared to Kitzbuehel, Bormio, and Beaver Creek. However, even the men do not want to race really hard courses like that every weekend.
But things were a LOT worse back in the 80's when some of the courses the women raced on were little more than glorified wax-test tracks barely over a minute long that were probably as boring for the women to ski as they were for spectators to watch. Thankfully, some of the people at the top woke out of their slumber and realized the women could handle tougher courses and that it would be worth more tv ratings if they did. The addition of harder, longer and more technical courses like some of the men's courses was one of those changes, although it appears they are doing it slowly and methodically. For instance, when the women first started racing Lake Louise, the organizers put in a new, extremely flat and easy part to bypass a section with jumps that they thought was too dangerous for the women, but eventually the women came to race the identical course that the men do.
Having said that, I agree with some of the other posters; strength is an issue and I doubt very many women have the kind of strength to race a course like Kitzbuehel, which has about half as many entrants as the other races, and yet still manages to put at least a couple of guys in the hospital every year.
Oh, and FWIW, men and women would have a hard time competing on even footing against each other because extra weight has a big effect on the outcome of downhill races, especially on the flatter and easier courses. Few of the women weigh as much as the average male DH racer.