Hi Julie,
Not sure what species of wood in your picture, it could be a dense heavy tropical hardwood like teak, ipe or such. In that case, they usually have naturally dense grain with natural oils which makes them okay for that purpose. Think of teak wood decks on boats which require continuous maintenance for looks only, otherwise they need no finishing. The Navy used teak for the decks of warships until they realized they burn.
That said, I built a "temporary" wood frame surround for a recycled stainless steel kitchen sink I found. I used cheap pine furring strips which is probably the worst possible wood for the purpose. I used a very thinned (mineral spirit) varnish which I repeatedly applied until it stopped being absorbed. It's been several years now for that "temporary" wood surround and it still looks like new. I don't mop up water spilled on the wood and it still looks like new. So much for "temporary"
The theory I used is that the thinned varnish would soak into the wood pores on each application so that virtually all the pores became polymerized varnish filled and thus virtually "plastic" and waterproof. It doesn't look plastic like the thick epoxy coating used on commercial bars and table.
Bottom line is that I'd do it for a "permanent" vanity like your picture. I does look cool (I have a "thing" for teak). If it were mine, I'd probably keep a towel handy to mop up any water to be sure. I'd use the repeated application of thinned varnish as a finish which looks like an oiled finish.
-- Teri