If you are seeing worms on your radishes, proper identification of the pest is the first step in controlling the situation. Two types of wormlike pests are known to bother radish plants.
Root maggots are small yellow to white wormlike pests that bore holes in the edible root of the radish. They live in the ground and are often not detected until the radish is picked and the damage is already done.
Green worms found wandering on the leaves of radish plants are likely the imported cabbageworm. You are more likely to see this pest on radishes planted near cabbage or broccoli.
To rid your garden of root maggots, dust the plant with diatomaceous earth. Mix the same into the ground surrounding your radish plants. If cabbageworm is the problem, remove the pests by hand and cover the plants with row covers to prevent the butterflies from laying their eggs on your plants.