Mix equal parts hydrogen peroxide and liquid dish detergent in a small bowl. Test the mixture in an inconspicuous place on your cabinet to make sure the peroxide will not bleach the paint.
Dampen a soft cloth with the mixture, and gently rub the area until the stain is removed.
Use another cloth dipped in water to rinse the cabinet. If the stain is still visible, make a paste of baking soda and mineral oil. The consistency needs to be rather thick, about the same thickness as peanut butter.
Spread the paste over the wine stain, using a clean cloth. Gently rub the paste into the stain. Brisk rubbing could scratch the finish.
Rinse with a damp cloth, and then dry the area with another cloth.