Absolutely dry them, if you put anything damp in your fly box the condensation will rust other flies. Here is a little trick to dry everything, open the boxes and put them into your oven overnight, assuming its gas and not electric, the pilot light is enough to apply dry heat and in the morning, everything is dry.
I do it all the time and it works very nicely. Or you can just open them in the sun all afternoon, I prefer to stick them in the oven right away when I get home. It drys the used flies and drys out boxes.