Byzantines are great against Mongols
Try an all-cav stack starting with fifteen of the best horse archers you can field.
(Skythions are not really up to it, but Byz Cav are surprisingly effective especially if you can get them an armour upgrade or two.)
Add a high dread general.
Then four of the best heavy cav you can find. Lancers will do, but try to field at least Latinkon and if you're playing Kingdoms: Crusades (I wasn't sure from your post) get some Archetopoloi out there, these bearded oldies really know how to skewer some Mongol.
Horse archer tactics are described in detail elsewhere (try Doug-thomson's excellent guide "Hunters all their Lives" somewhere on this forum), but generally speaking try to surround them and take out their missile cav, foot missiles, heavy cav in that order
don't let their general escape the battlefield - if you can kill him that's one step closer to eradicating the mongols
good luck soldier
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