Watching "The Two Towers", specifically the battle of Helms Deep, made me think of this. I would imagine that water would be horrible for the bows (the strings and bindings and wood) and the arrows. Was it not a problem? If it was, how did they deal with it? Were there precautions they took? Would some sort of fat/oil prevent all water damage?
Rain would hardly be ideal for medieval archers but it wouldn't be the end of the world for them either. Longbows and wooden crossbows would be made out of a single piece of wood so they wouldn't have to worry about any glue coming undone and rendering the weapon useless. Steel Crossbows similarly wouldn't be worried about a little water since it's not like rust could set in over the course of a battle. Composite Crossbows would be at the greatest risk since most medieval glue could come undone in rain but composite crossbows were wrapped in tree bark to protect the actual bow section. This wouldn't necessarily keep the crossbow safe if you submerged it in a deep pool of water or anything but it was easily sufficient for rain.
When it comes to strings the evidence is a lot less clear. As someone else already noted modern archers put a layer of wax on their strings to keep them safe. Unfortunately we don't really know how far back this practice goes. There's some evidence to suggest that by the 15th century archers would put fat on their strings to keep them dry but we have no idea if that was standard practice earlier. Bow strings are unfortunately a bit of a mystery. We know they were made of Hemp, Silk or sometimes linen but they're not often talked about in medieval texts and they pretty much never survive.
Arrows certainly wouldn't appreciate being rained on. With the prevalence of water soluble glue in the Middle Ages the fletching would probably suffer quite a bit if it was rained on. That said Fletching was also often tied on with string as well as glue which could hold it together well enough for the arrow to reach it's target. Unfortunately it's hard to know what the standards were for the craftsmanship of the average medieval arrow. Probably not very high, though. We know the English made arrows from Poplar in the Hundred Years War which is a pretty terrible wood for shooting but is really easy to make arrows out of.
Overall, though, it should be said that medieval battles weren't fought in the rain very often. The rain would turn the ground into mud and fighting in a giant muddy field was not something people generally seemed to want. Battles were often preceded by a lot of positioning and eyeing up the other army. These armies generally camped within a few miles of each other and several days could pass before they actually fought. Armies could get up, form battle lines and wait for hours with no one attacking. This generally meant that you could wait out the weather most of the time.
For more on the subject of medieval archery I'd recommend Jim Bradbury's The Medieval Archer. It's probably the best work on the subject. For crossbows more specifically Josef Alm's European Crossbows: A survey is amazing if you can find a copy (good luck!). Jean Froissart's Chronicles offers a good primary source account of a couple of campaigns which featured some terrible weather.
At the Battle of Crecy in 1346 it rained quite heavily prior to the fighting (the reason the French Cavalry charge was slowed in thick mud).
The battle started with the Genoese crossbowmen who were fighting for France advancing in the van and opening fire. However, prior to the first shots, they were rained upon and were unable to keep the equipment dry. Due to the high tension of a crossbow it was impossible to remove the string (in the field) and protect it from the damp and thus their crossbows got wet.
When the Genoese actually opened fire they had forgotten to get closer due to being hampered by wet strings and thus nearly the entire volley fell short. Other factors like the sun in their eyes would have also had an effect but wet equipment is believed to be one of the main reasons.
The English and Welsh bowman in the meantime had taken off their bow strings and put them under their hats or in bags while they got rained upon. When the rain stopped they put their strings back on and were able to fire their bows unimpeded into the French ranks.
The Genoese were placed under such a heavy barrage that they retreated, only for the French Knights to ride them down as they charged the English.
In terms of fighting in constant rain I do not know what would have been a solution.
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