Sand your stained or painted dresser completely with medium-grit sandpaper to remove any lumps or bumps and old varnish from the exterior. If you are working with an old stain or paint job, consider completely removing the old finish and applying a new one, as a varnish coat is likely to peel off an older coat of paint or stain. Sand the coat in small circles until the finish is smooth.
Wipe down the entire surface with a dry cloth to remove dust and vacuum the area. Fill a bucket with cold water and wet your cloth in it; wring the cloth to remove excess water. Rub down the dresser with the damp cloth to remove small particles of dust and residue. Let the dresser dry completely and repeat once or twice until all of the residue has been removed.
Open a can of water-based polyurethane and mix slowly and carefully with a mixing stick. Pour a small amount of the polyurethane into a small paint bowl or receptacle.
Dip your paintbrush into the polyurethane and apply it to the dresser in long, slow strokes with the grain of the wood. Apply one thin coat of the polyurethane to the dresser and let it dry completely, which should take only a few hours. The coat will appear slightly milky until it is completely dry.
Sand the coat lightly with the fine sandpaper after it is dry to smooth the coat out and remove any brushstroke marks. Apply several more coats until you have achieved the level of shine you desire, allowing each coat to dry and sanding between each application carefully.