Varnish would be your best bet. It that all real wood or are some of the panels MDF? I can tell from the photos.
Give it a couple days with good ventilation to gas out. If you could put it outside in the sun that will help too.
You honestly only need to seal the bottom and maybe 3 or 4 inches up the sides, those are the only places you will really get water splashes and poo paintings.
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