The white refrigerator in your kitchen didn't come with yellow handles, but they're yellow now. Even after cleaning, they'll still look pretty grungy against the bright white porcelain cabinet. New handles are pretty pricey, so why not just paint them? It's easy and cheap. Buy a can of paint formulated to paint plastic (Rust-Oleum Specialty Paint for Plastic is one choice) at a home center. Then pick up a bottle of degreaser and a scrubbing pad.
Remove the handles (Photo 1). Then wash them with degreaser and hot water. Use a scrub pad on textured areas to remove the dirt. Pay particular attention to dirt in crevices. Then wipe the handles with a dry cloth and let them air-dry.
Mask off any emblems or chrome trim and paint the handles (Photo 2). Paint the vanity caps at the same time. Then reinstall the handles. You may have to repaint them in a few years. But in the meantime, they'll look a lot better.