It looks like particle board under your veneer and once that swells there is no going back. You might be able to trim of the bottom part of the wall and cover the gap with wood trim finished on all 6 sides.

Only before they are installed. A brass or stainless foot type detail that elevates the side panel off the floor would have been ideal to design into the installation.

Is there something like a brass clamp that could be applied on top that wraps around?

It would have been nice for the kitchen cabinet company to advise us of this before we spent an insance amount of money on these cabinets.

Real wood would have been even more insane. What you can do if you like brass, or stainless steel or other metal available in thin sheet is have a sheet metal fabrication house fab a u-shaped piece to fit over the bottom few inches of the veneer. Extending it all over the kitchen cabinet base areas would be expensive, so only the protruding portions might be treated. You could even use imitation wood Contact paper to disguise it.

Cloth bath mats might be useful to keep puddles from forming.