DIAGNOSIS The Clean indicator light is tied to a sensor that estimates soil levels. If it stays on constantly, it's not broken, but it's likely stuck in a 'high soil detected' state, often because the sensor is dirty or the filter needs cleaning.
FIX The first step is to clean the filters. Twist and lift out the upper filter assembly (the fine mesh cylinder). Then remove the flat coarse filter underneath. Rinse both thoroughly under hot water, using a soft brush to remove any grease or debris. Also, wipe around the edges of the sump where the filters sit. Reassemble them correctly—the coarse filter lies flat, then the fine cylinder twists into place. Run a hot water cycle with a dishwasher cleaner or a cup of vinegar placed upright on the top rack.
IF THAT DIDN'T WORK If the light persists after a filter cleaning and a cleaner cycle, the turbidity sensor (the little window usually near the sump) might be permanently filmed over. You can try gently cleaning that small glass or plastic sensor window with a cotton swab and rubbing alcohol, being very careful not to scratch it.
HEADS UP Don't ignore a constantly lit Clean light. It can cause the dishwasher to use more water and energy by defaulting to heavier soil settings, thinking every load is extra dirty.