Water trapped in bilge

[Mariner]

For as long as we've had our 36' Express Trawler, there's been water that accumulates outboard of the longitudinal hull members, running the length of the hull. It's usually not much...just an inch or so...no more than a couple gallons in the whole boat, but it's enough that it keeps it humid and musty in the bilge and engine room in the winter. It's generally freshwater...rain that finds it's way into the bilge from the deck here and there. The problem is that it promotes mildew growth in these spaces and limits options for storage. It's generally too shallow to evacuate with a bilge pump, and besides, there's probably five or six such spaces throughout the boat...it'd be ridiculous to install bilge pumps in all of them. It could be easily solved with a few well-placed sprue holes through the longitudinals, allowing the water to simply drain to the sump in the keel, but I'm not inclined to be drilling holes in those structural members, especially without knowing what's in them.

Anyone got any ideas?