The other problem, besides rain that doesn't drain, is that birds like to make nests in those boxes!
I made covers out of 1/2" StarBoard, with a lip on the top so they rest on the edge of the coaming, and with two eyes attached to the lip anchoring bungee cord that wraps around the nearby cleats so the covers are held in place. The bungee cord lasts about two seasons.* The StarBoard has lasted...gee...22 years so far?
And years ago I dug out the old warped plywood layers and faired over what was left with white-tinted epoxy.
*The bungee cord system replaced wooden cleats I'd put inside the covers. One cleat was fixed in place; the other was attached to a toilet-paper-roll-holder anchored to the cover, which worked as a spring-loaded piston to squeeze the cover against the hole. This setup was greatly amusing, and worked pretty well, except that for some reason they don't make salt-water-resistant toilet-paper-roll-holders, and they only lasted a few weeks before the springs rusted.