Make pesto: Bring 2 cups water to a simmer in a small saucepan. Add basil, and submerge with a spoon; immediately drain in a colander. In a blender, combine basil, pecans, garlic, lemon juice, oil, 1/4 teaspoon salt, 1/8 teaspoon pepper, and 1/4 cup water; blend until smooth. Set aside.
Make salad: In a large saucepan, cover potatoes with salted water by 1 inch. Bring to a boil; reduce heat, and simmer until tender when pierced with the tip of a paring knife, about 15 minutes. Remove with a slotted spoon (reserve pan of water). When cool enough to handle, cut potatoes into quarters; place in a large bowl.
While potatoes are cooling, return reserved water to a boil. Add green beans; cook until crisp-tender, 4 to 6 minutes. Drain well; transfer to bowl with potatoes.Add pesto, and toss to coat. Season with salt and pepper; serve immediately.