Paint or refinish in clear coat would depend somewhat on why it's been getting wet. Are you going to raise it or alter it somehow to keep it from getting wet along the bottom boards? If bar stools will be used, having a sturdy finish on the kick area is good.

For some reason I can picture this in a dark brown or even black.. but the countertop makes me feel like white.

I like the wood. If you want some color, add a aqua or blue stain.

Add color with stain

Or leave it natural and add color with your stools