Keith Newton
I wouldn't bother with a finish for what you desire. The BLO, will slow absorption a little, but will circle so badly, that it will look worse than no finish at all.

However, none of the woods you mentioned will absorb moisture as fast as a frozen mug can produce it, so you might want to rethink your expectations.
Besides, if they are dry and flat when you make them, they will cup when they become wet, producing a little rocker for your mug.

Maybe a better solution, would be to make little grates, which allow the moisture to just flow through, could cut down on the drips, but lifting and tilting will likely still allow some drops to fall.