What did a typical battle between a Crusader army and a Muslim army look like? How did the crusader armies deal with horse archers?

There is not "the typical battle" during a series of conflicts that spanned hundreds of years. Tactics, equipment and terrain were constantly changing.

However, there are some general principles. Crusaders would generally try to keep their guard up. Horse archers could strike any time, so crusaders would travel in formations, with armor and weapons ready. Or at least that is what successful crusaders did, e.g. Richard I during his march from Acre.

Also, even in the heat of battle the crusaders would try to keep formation (again, at least the successful crusaders did that). Horse archers excel at feigned retreats and the 1st and 2nd crusade were warned by the Byzantine Emperor not to fall for that ruse. E.g. if the knights charge and disperse, they are easy victims. If they keep their discipline and do not charge or if they at least charge as a big group, they have a better chance of survival.

Speaking of cavalry: Crusaders/Latins who encountered Muslims would sometimes just damn the odds and charge instantly. This might seem rash, but it worked surprisingly often, e.g. at the battle of Mont Gisard, where King Balduin and Reynald de Chatillon defeated Saladin. A possible explanation is that this allowed the Latins to take their enemies by surprise.

Also "Combined arms": Asbridge quotes one Muslim describing the Christian infantry during the siege of Acre: "Like a wall behind their mantlets, shields and lances, with levelled crossbows, refusing to break formation". Effectively, the horse archers could circle all they liked, their arrows could not penetrate the Latins' shields. Neither could they charge home against massed spears and men and the crossbowmen, standing protected by the shields, would inexorably pick them off. Similarly, at the battle of Dorylaeum during the 1st Crusade the Turkish horse archers did not prove effective against heavily armored Franks. The prerequisite of this tactic, obviously, was the Latins' discipline not to break ranks no matter what and this often proved difficult to achieve. Also a part of combined arms is that the Latins would just hire mercenary horse archers to fight for them, rather than against them.

That being said, most military actions would be sieges and raids, rather than pitched battles (although the latter are of course much more famous). However, during these the tactics were not that different. Raiders tried not to fight at all and during sieges both sides tried to starve the opponent or dig tunnels, build catapults, etc. ... A difference is that the Christians usually ruled the sea (Italians) while the Muslims sometimes could burn Latin siege engines (and soldiers) with Greek Fire. And of course, sometimes there was no war at all (except for inevitable small-scale border raids), when both sides had signed a ceasefire.

Source: Mostly based on Asbridge: The Crusades.