Set aside one whole radish for garnish. Thinly slice five of the radishes. Cut remaining four radishes into halves or quarters.
Use a vegetable peeler to cut thin, ribbonlike lengthwise slices from cucumbers, discarding first slice. (You should have about 2 cups.)
In a bowl combine radish slices, halves, and quarters; cucumber ribbons; jalapeno pepper; basil, sugar; and salt. Drizzle with olive oil and white wine vinegar. Toss gently to coat. Garnish with reserved whole radish.