Last time I used Khergit Horse Archer everyone ran into the city and hid because they were sick and tired of getting circled (non-shielded infantry) or run over (archers with their bow out), good times.
I usually never die before I use all 90+ arrows because the cavalry can't catch me alone (edit: Also chasing a horse in a large blob won't work either! spread out), I'm not stupid enough to run into the infantry and the archers there needed to learn that a smart guy on a horse won't keep going the same way at the same speed.
As for stopping horse archers my advice is..
Infantry = shields, personally I don't even BOTHER if you have a shield, I'll just go look for someone else to harass or shoot you in the back when your busy fighting. As long as you're on foot w/o ranged though you won't be catching me any time soon, throwing weapon maybe? I've gotten an axe in a horsemans face before so it's definitely possible.
Cavalry, attack in at least a group of 2, take a side and try to stop them from turning off that way, if you have a lance you can just couch their horse if they try to turn and then they're royally screwed. If you have a bow then the solution should be fairly obvious.
As for archers, if your aim is good enough it should be a quick finish, if not then try to take out their horse but do keep in mind a decent horse archer will be WATCHING YOU (and likely shooting you) if they know you're there so try to take out the horse before they spot you, if they start moving erratically then they likely know you're aiming at them.
Lastly, taking a horse archer by surprise is always an option, especially for archers. If you find a nice spot it'll turn the tables and you'll become the harasser that THEY want gone.
^ All of the above, fairly obvious stuff that also works for me, I usually go with archer though so catching a Khergit isn't a problem.