Elbow grease and brass polish. Repeat as long as you keep dripping water on them.

Metal polish applied to the underneath part first, to see if it removed the dark finish. If that goes well, apply metal polish only to the deteriorated areas, then paint with clear polyurethane. Two thin coats are better than one thick coat. You may have to apply a little dark "oil rubbed bronze" or black touch up paint, too. Rustoleum's paint products for metal surfaces are excellent, IME. Tiny minor dings (a match head, not a dime) can sometimes be touched up with black Sharpie or black nail polish.

Alternatively, if the pulls don't look great after some effort, trade the sink pulls for the over-fridge pulls or pulls on ankle height bottom drawers. Bottom drawer pulls are too low for tiny flaws to be seen clearly and they are often used quite infrequently, so their finishes aren't worn out quickly.