How would you protect a plywood container from moisture and abrasion?
The usual approach is a couple coats of polyurethane varnish. I'd sand it a bit to clean it up and give it a coat on all exposed surfaces. Steel wool between coats. It'll be apparent when you've applied enough coats to seal the edge grain well enough to protect it. Reapply every couple years as needed. 

If you want the plywood to have a color, simply stain or paint it first. If you use a good oil-based paint (like a floor paint), you could omit the varnish and use two or more coats of paint. Otherwise paint with a flat sheen, then varnish. Glossy sheens would require light sanding for good bonding of the urethane.