Blend the basil, vinegar, 1 teaspoon of salt, and 1/2 teaspoon of pepper in a blender. With the machine running, gradually blend in the oil.
Place the arugula and radicchio in a wide shallow bowl. Using a vegetable peeler, shave the carrot over the salad. Shave the cucumber into a medium bowl. Pat the cucumber shavings with paper towels to absorb the excess moisture. Add the cucumbers to the salad.
Do-Ahead Tip: The vinaigrette and salad can be prepared 1 day ahead. Cover separately and refrigerate. Whisk the dressing to blend before using.
Toss the salad with enough dressing to coat. Season the salad, to taste, with salt and pepper, and serve.
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