If your wash stand is French Polished, which it should be, this will provide an excellent base for painting, you will be best to de-grease it first because it has probably been waxed at some time in its life. Giving it a scrub with some white spirit and a scotch pad ( the green pan scourers are good to use ) will de-grease it and key it up in one operation, but make sure it is good and dry before painting. You could use water or oil based paint on top, it all depends on the finish you are after.
One thing you might want to do before embarking on this is to check its value. Nearly all wash stands have devalued a great deal like most 'Brown" furniture but there are a few exceptions, It would be a pity to paint it only to find it is a 'Gillows' piece! You could always send me a photo of it first if you have any doubts.