How to finish bathroom vanity junction with the floor?
If you can mark the shims at the face of the cabinet, remove them temporarily, cut them slightly shorter than the mark, use a black marker to "paint the visible sides of the shims and set them back in about a 1/8" behind the face of the cabinet. Using a tube of clear water cleanup caulk, with the tip cut really small, say about 1/8" at the tip, draw a line around the bottom of the cabinet, filling the gap. Wipe the excess flush with the edges of the gap, leaving nothing on the adjacent surfaces of the cabinet or tile, the residue will show up in the glare from the lights in the bathroom.

FWIW, sealing the bottom with caulk may be good if the sink, waterlines or drain does not leak in the years to come. I would do a little here and there with the clear caulk at the escutcheons to keep the potential leaks in visible places, rather than running under a cabinet and collecting there. If the toilet floods, it is a good thing to caulk there, it is a bad thing if the drain or something else at the inside of the cabinet leaks. That is why I suggest the caulk here and there wiped with a damp sponge or cloth to keep the caulk in the joints only, and force a leak to show itself.

I see you have no base there yet, I would caulk the gap at the wall to the floor before you install the base. Just my 2 cents worth...