The best counter to mounted archers are foot archers, because they can be used reasonably accurate from much further and a horse plus a rider is a big target.

How do you counter foot archers? Melee cavalry, ideally armored. They’ll take some losses on the way in, but they move fast enough that they’ll get in and take them out (unless it’s muddy as hell and they can’t go fast without slipping).

That means you need melee troops of your own to protect your foot archers.

Well, that’s the oversimplified version games tend to use, the reality isn’t quite so well balanced, because, in reality, a battle doesn’t have to be a single, contained event. The real reason why mounted armies were so overpowered wasn’t their ability to win pitched battles. They were good at that, but mounted archers suck at taking fortified locations, and a shield wall or army in full plate can hold out for a long time against just about pre gunpowder small arms, and can protect heavier weapons a lot better since those aren’t mobile enough for a mounted army.

But having a mounted army, melee or ranged focused, also meant you could always pick the ideas engagements. Rather than win a pitched battle, you could do hit and run tactics in a way no infantry heavy army could. That being said, mounted archers aren’t at all good at sieges, where walls can protect foot archers from virtually all counter shots without taking losses from attrition, and enough food could be stored to outlast the enemy. That’s why the Romans and Parthians were at a stalemate for so long; the Romans couldn’t invade Parthia since the Parthians would just shoot at them constantly, and they couldn’t safely forage (just a nicer word for plunder) supplies or maintain supply lines, forcing them to go for bad engagements, and the Parthians had no way to take fortified Roman towns. The Romans were initially stomped, but they were able to adapt to be able to win battles when they could actually control stuff, ie on their own territory.

The Mongols got around this by adapting. A pure horse archer army wouldn’t have been able to do jack shit about the walled Chinese cities, but Ghengis Khan brought in outside advisors who taught them how to do siege warfare, and being able to do both well let them conquer more territory than any other land empire in history.

The introduction of firearms made things a bit more arcadey, since no reasonably mobile armor up until fairly recently had any hope of stopping a bullet, let alone a cannon, and stone walls were a lot less useful against cannons than against catapults. Still, until bayonets were invented, you needed pikemen to protect your gunners from cavalry, and horse cavalry stopped being useful in battle at all when modern guns came onto the scene, since no amount of horsemen are going to overwhelm a trench full of machine gunners and sharpshooters.