I finish gunstocks with blo. My shotgun is finished with it and I use it heavily for competition rain or shine. It does require some mantainence so I'm not sure I would leave it as the final finish for a bathroom vanity.
The old rule for guns goes something like One a day for a week, Once a week for a month, once a month for a year and once a year for a lifetime.
I do a grain fill by wetsanding with a drop or two of blo. You have to work your way up the grits starting with 220. it will leave a mud coat that you rub in good each time and let it dry. by the time you get up to 800 your peice is like glass and has and incredible luster.
I have used it as a figue enhancer as well. If it is a hunting gun and I know it is going to be out it poor weather all the time. Sometimes blo will cloud up when it gets rained on but it will dry out again. Watco dannish oil can give a similar finish and seals the wood a little better.
If it were my cabunet I would wetsand with blo untill tou are happy and then let it cure for a month and top coat it like the other fellas mentioned. a bathroom can be pretty moist (especially if there are girls in the house). And you have other things to do than constantly maintaining your vanity (especially if there are girls in the house).