How to fix stinky bathroom sink?
I'll echo the comment and say that you should make sure you have a proper P-trap installed under the sink. This trap holds water and provides a seal against sewer gases getting up into the bathroom. Without one, gases will leak in constantly, and will be displaced by water down the drain which can force the gases up into the bathroom even if normally it's not detectable.

I'll also ask if there is a proper vent for this sink. Even with a P-trap, if there's nowhere for back-flowing gases to go, they'll bubble up past the trap.

One more thing it could be is the overflow drain. Depending on the design of sink, the overflow can hold a small amount of water at the bottom where it tees in to the main drain, which can become stagnant. Run water down the drain and you'll force some air up the overflow (to make way for the water coming down), which will have that stagnant smell. To diagnose this, plug the sink and begin filling it; you shouldn't get any musty smell at first because there's no air movement. Once the water level hits the overflow drain, you will start smelling the musty smell for a while because the water is displacing the gas, which wants to rise above the water and so will move up into the bathroom. If this is the problem, you can ameliorate it with some foaming pipe snake; pour it down the overflow drain and it will clean out any caked-on gunk which contributes to the smell, and which may be trapping the water. The real fix is to make sure there's no "damming" effect of construction defects at the bottom of the overflow drain (a lip of porcelain, issues where the overflow meets the metal drain downpipe, etc).