Waterproof paint for the flat parts, tape for the edges.
Any kind of spreadable waterproof fluid will do for the large faces of the cardboard sheets. Paint or glue could both work, but paint is probably more cost-effective. There are many types of waterproof paint (acrylic, enamel, anything listed as exterior-grade).
The edges will be harder to deal with using paint. Even if you managed to paint the edge itself successfully, water could still trickle down the corrugated part to the inner parts where your paintbrush couldn't reach and destroy the cardboard from the inside out. But the edges can be covered using waterproof tape. Duct tape would be best for strength and durability, but if you don't want it to be visible and spoil the image of your painted props, you could use a strong clear tape instead.
See also this Quora article:
There are various ways you could seal cardboard but basically you'll have to cover every part of it that would ever come in contact with water or even humidity, and do that completely and without even tiny gaps.
So you could seal it with waterproof (or at least water-resistant) liquids, or with waterproof tapes (duct tape, clear packing tape, etc), or by using something like shrink-wrap plastic or other solid but flexible plastics, or with flexible metals like aluminum foil, or even just by enclosing it inside a waterproof acrylic or glass display box or dome.
The most water"proof" liquids would be permanent paints (acrylics, latex, enamels) or clear finishes like polyurethane and epoxy resins and even less waterproof ones like thinned-down PVA glues** and acrylic mediums, etc.
(Many other pasty, etc, materials that are thicker or not intended as finishes/paints could work too but they wouldn't usually be as smooth or thin.)
** aka permanent white glues, craft glues, etc --Elmer's GlueAll is most commonly used or book binding glue-- and "decoupage mediums" like Mod Podge which are already thinned down white glues