A clean battery corrosion makes a bigger difference than you'd think. Here's the most effective method, plus the mistakes most people make.
Wear rubber gloves. Carefully remove the corroded batteries. Avoid touching the white powder -- it is caustic.
Dip a cotton swab in white vinegar or lemon juice. The acid neutralizes the alkaline corrosion.
Gently scrub the battery contacts with the cotton swab and an old toothbrush. The corrosion should dissolve.
If corrosion remains, make a paste of baking soda and water. Apply, let sit 5 minutes, then scrub.
Wipe everything dry with paper towels. Make sure no moisture remains before installing fresh batteries.