If I remember right there is a lip that stands up around the perimeter of the opening. This lip is under the lid when closed.
You could takes some modeling clay and use it to determine the clearance between he lip and the lid. Once you have that size you can source automotive trunk weatherstripping that can be slid on the lip and will have a bulb seal top. It comes in a plethora of sizes. I think Alan even posted an on line company that sells it a while back, but I don't have the link.
This should make the drawer water resistant. I do not think there is a way to make them watertight beyond welding them closed.