I own a number of wood tables and none of them are finished on the underside. Do it if you like, but I think it's unnecessary work.
As far as water getting through the slats, I would be more concerned about dampness between the slats than water on the underside.
A table as solid as that one looks, I don't think you need to worry about water causing any structural damage. By the time that thing rots, we'll all be too old to lift kettles.
Unless you plan to leave it outside in the elements. That would be a whole different story.