Preheat oven to 375 degrees. Place walnuts on a rimmed baking sheet. Toast in oven until fragrant, about 8 minutes. Let cool slightly. Finely chop, and set aside.
Whisk together vinegar, yogurt, mustard, 1/2 teaspoon salt, and the toasted nuts in a small bowl; season with pepper. Add oil in a slow, steady stream, whisking until emulsified. Set dressing aside.
Bring a medium saucepan of water to a boil. Add potatoes, and cook until tender, about 10 minutes. Using a slotted spoon, transfer potatoes to a colander to drain. Set aside.
Prepare an ice-water bath; set aside. Return pan of water to a boil. Add green beans, and cook until tender and bright green, about 4 minutes. Using a slotted spoon, transfer to ice-water bath to stop the cooking. Transfer to a cutting board, and cut into 2-inch pieces.
Arrange arugula, potatoes, and green beans on a platter. Season with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Drizzle with dressing; toss to coat.