Peel cucumber. Use the vegetable peeler to make strips of cucumber until you reach the seeds all the way around. Discard peel and middle of cucumber. Place cucumber strips in a bowl.
Peel mango. Use the vegetable peeler to make strips of mango until you reach the pit all the way around. Discard peel and pit of mango. Place mango strips in the bowl with the cucumber.
Combine orange juice, lemon juice, vinegar, basil, salt, and black pepper in a blender and puree. Drizzle oil in slowly while blender is running. Mix dressing into the bowl with the cucumber and mango. Cover and refrigerate at least 2 hours.
