According to my GGM and GM (both long deceased) women who rode horses "in the day" either wore pants or split skirts (e.g. http://www.gentlemansemporium.com/store/000981.php).

As a kid we had horses. We had an old sidesaddle. If the cinch is tight so the saddle can't rotate, they are actually fairly stable. your left foot is in the stirrup, your right leg rests in a "hook shaped" horn. No, I didn't wear a split skirt, only jeans.