Prepare a gas grill for direct-heat cooking over medium heat; see Grilling Procedure .
Whisk together dressing ingredients in a bowl.
Cover potatoes with water in a 4-quart pot and season well with salt. Bring to a boil, then reduce heat and simmer, uncovered, until just tender, 15 to 20 minutes.
While potatoes cook, season salmon with 1/2 teaspoon salt and 1/4 teaspoon pepper. Oil grill rack, then grill salmon, covered, turning once, until just cooked through, 8 to 10 minutes total. Cool slightly, then gently break into large flakes. If desired, reserve skin for crumbling over salad.
Transfer potatoes with a slotted spoon to a bowl. Add green beans to boiling water and cook, uncovered, until crisp-tender, 4 to 5 minutes. Drain and transfer to an ice bath to stop cooking.
Halve potatoes while still warm and toss with 2 tablespoons dressing.
Toss green beans, cherry tomatoes, and arugula with enough dressing to coat, then toss with potatoes. Divide among plates with salmon and eggs. Sprinkle with basil (and salmon skin, if using). Serve with lemon wedges and remaining dressing.
Cooks' note: Salmon can be cooked in an oiled hot grill pan (uncovered) or in a hot skillet with 1 tablespoon olive oil over medium-high heat.