Put the potatoes in a large pot; add water to cover by 2 inches. Cover the pot and bring to boil over medium-high heat. Cook until the potatoes are cooked through, about 30 minutes. Drain and let cool slightly.
Meanwhile, bring another large saucepan of water to a boil; salt generously. Blanch the green beans until crisp-tender, 2 to 3 minutes. Drain and then shock in an ice water bath to stop cooking.
When cool enough to handle, cut the potatoes into 1 1/2-inch chunks and place in a large serving bowl. Add the green beans, red onions and tomatoes and toss together. In a small bowl, whisk together the vinegar, olive oil and dried basil. Pour about three-quarters of the dressing over the salad and toss to combine. (The potatoes should break up a bit and coat the salad.) Add salt and pepper to taste. Serve at room temperature, garnished with torn basil.