It's also wet now.
What kind of caulking and can you clear it out well?
Assuming it's not leaking through the painted plywood, but leaking at one or more points along seams
I'd be inclined to use a nuke on this. Clean the caulk out, scrape the paint off in the joints. Then PL Premium construction adhesive is your new friend; it doesn't mind if things are damp (in fact, it needs to absorb moisture in order to cure quickly/properly). Two ways:
- one - pretend the PL is caulking and fillet the joints; or
- two - glue in a piece of quarter round (or other strip of wood), covering all the seams and extending past where the caulking was.
With this one, you're going past the point where the leak is/was, and past any old caulking that could prevent a bond. This depends on getting the elastromeric paint off so you're bonding wood to wood. The PL seals these joints too. Wouldn't hurt to us PL to fillet the quarter round to the sides/bottom either.
If you constructed the box with a joining strip of wood at the seams, then use something aggressive to remove the caulking and paint so you've got bare wood, then fillet it with PL.
p.s. don't get PL on your hands. If not cleaned off immediately, it will be there until the underlying skin comes off.
If you find it's leaking at a specific point, remember that Gorilla Glue (not their "wood" PVA glue), like PL, needs moisture to cure. I had a leak in my plywood trailer top, coated with elastromeric. There was a hidden void in the plywood right at a butt-joint that leaked, so water had a path through. Just injected the Gorilla Glue into the wet voids and wet seam. Sealed right up as the glue absorbed the moisture to cure, expanding slightly too.
If for some reason it's the elastromeric paint on the plywood that is leaking, then it may or may not touch up with more paint. Or you could get it off and try epoxy now. Only it may be wet and you're running out of time. Another nuke solution is to remove the paint and use a putty knife to put a thin layer of PL into/on the scraped wood. It won't care about the moisture.