There are a few things you can do to your bow case that helps draw moisture away from your bow. Silica packs used to keep metal tools dry work good. Some fill a sock with dry rice, you can dry both in the microwave. I've even seen a guy who lines his bow case with baby diapers and maxi pads. Most bow cases are lined with foam that will store the moisture next to your bow. Anything that you can do to draw moisture away from your bow will help while your bow is in the case. A good wicking material between your bow and foam helps a lot. Your bow case should suck the water off your bow, move water away from your bow and keep water away until you can open it up and dry everything out. Once home, I get my case open so everything can dry.
It seems to me the parts that rust the fastest are usually the screw heads on the bow. Any rusty screw I find, I remove, clean, and use a good rust converter and then paint black or green. A little finger full of bow wax over each screw also helps. The next thing that rusts fast is the tips of my broad heads. My quivers have foam that holds moisture, got to get the bh's out of the foam and make sure everything dries good too. If your bh tips rust, sharpen the tip, and put a drop or two of WD40 on just the tip. An old guy I used to know sharpened ever Thunderhead broad head tip, and blued them with gun bluing but a drop of WD40 works for me.
It's also not just rain and snow you need to be aware of. Taking a bow or gun in and out when it's cold is tough on equipment. A cold bow or gun brought inside will draw and condense moisture. Fog up. Even inside the case. I try to avoid bringing my bow inside when it's cold and I'm hunting every day and also get my bow out of my truck when it's warm. The extreme temperatures inside a vehicle isn't real good for your bow either. 120 degrees will melt the wax right off your string. Anytime you cook a bow in a vehicle, you need to re wax everything and check that anything glued like mole skin, plastic and rubber didn't come loose due to extreme heat.
I think that's about all I can add to what's been posted already except that if you are hunting in rain, I hope you practiced in rain. I know too many people who haven't mad a practice shot since opening day. Been carrying a bow around for 2 months and haven't shot it. With the weather changing daily, a bow is like a race track, you got to keep up with the changes. Up a tree, over the big one, is not the place you want to discover a squeak in your bow or that your bow shoots 2" higher at 20 degrees temp than 80 degrees temp. or your range is cut in half by a driving rain, even fog can take 5 yards off your max range. Shooting today, is the only way to be sure how you are going to shoot today.
American by birth, hunter by choice.