It's pretty new that horses are primarily ridden for fun. In the past they were used as a means of transport (carriage or ridden), for work (stock work etc) and in battle. In the past men used to work and be involved in these things (war, stockwork) more than women. The disciplines we have today often evolved from things that were a necessity in the past.
Most western riding is based on what riders had to do to work stock and a farm, and has been further developed to be what it is today. Dressage used to be training for the battlefield. Obviously, back then, men had the advantage, the skill and means to be involved in this. Remember, in the past men dominated everything, they were the "active" gender as opposed to the "passive" one. They were the ones that did everything.
Now it's a hobby. It does seem that, especially at the lower levels, there are more women than me. I don't know why, maybe because it's no longer such a status thing and it takes an enormous amount of time and money to keep horses. You have to really love it. I do find that there are more men competing in the western type sports, perhaps it's maybe because it's still part of their work, or they find it more tangible.
Horses were as much used in harness as they were ridden and there were women 'warriors like Boadicea who is said to have driven a chariot into battle with spikes on the wheels as far back as AD61
Women aren't really associated with fighting in battles but they rode for pleasure and transport and though we think of them mostly riding sidesaddle for respectability and modesty there are images from Mediaeval times of quite affluent looking women riding astride like this one
There's some evidence that women fought as warriors among the Scythians (ca 600 BC) and other nomadic peoples east of the Black Sea. But even they were relative latecomers, as horses were likely domesticated sometime between 2000 and 4000 BC.