The key thing with horse archers is initiative. Horse archers aren't that good at ranged combat and usually don't do much in a melee, but they are fast and can strike where you are vulnerable.

Horse archer weaknesses
1) Because they are not very good in ranged combat (lower accuracy, usually little armour, horse = big target, unwieldy) a ranged duel will dispatch them really quick. That you should ensure that your ranged units are well protected is self evident. But a good commander can use the horse archer's speed to make this difficult. The AI will not get into a ranged duel with horse archers, barring special circumstances.

2) They are very bad at melee. Plain light cavalry can beat them. However, you have the same problem: their speed. It is slightly lower than that of most light cavalry, but enough to help them get away. You need to trap them. However, here is a factor that aids you:

3) Their poor morale. It takes little to rout them of the field: a few casualties and a nice flanking should do the trick. They will usually rally and regroup, but by then they are tired and frightened, so it should take even less to rout them again.

They are a nuisance, and it requires proper equipment and some attention to get rid of them. But with that, they won't do much damage as long as you keep some pressure onto them.

If you lack the equipment, you might want to use the woods. It will hinder your archers too, but it renders their arrows almost ineffective.