I try not to hunt in the rain, as I just don't like to get wet. As a younger guy, it didn't matter. Rain never bothered wooden arrow shafts at all. Fletching needed to be treated to keep them from becoming flat.
One interesting thing I did notice once had to do with tuning. I was shooting arrows that were marginally tuned that shot well while dry, and a friend was shooting arrows that were bare shaft tuned. When feathers got wet, my arrows flew erratically while his remained very flat shooting. I wish I had arrows with goose feathers as described by Shadowhunter.