An acrylic latex paint in an enamel finish! Works well for cabinetry, wears, well, washes easily. Prime first!
Take one of the drawers to the paint department of the home improvement store and ask for their advice on paints and application tools. They have the latest products and know-how to do a professional job.
In a bathroom due to Steam and moisture, you cannot go with water base. It will crack and bubble. Thus your best bet is to go with:
A. Oil Based Interior
B. Any Exterior.
I recommend B.
I used an acrylic latex paint in an enamel finish. Eight years later, living through 3 teenagers, the vanity was in great shape--so the paint lasted and there was no moisture damage--even though the children would take long hot showers until the water heater ran cold! I changed the vanity after 30 years just because I was sick of looking at the same vanity! The paint was still fine--it sold on the first day at the Restore.
Here is how I updated my bathroom vanity... hope it helps! https://www.homeroad.net/2018/03/repairing-water-damaged-bathroom-vanity.html