Scrub the potatoes clean. Leaving the skins on, place the potatoes in a large pot. Fill the pot with enough water so that the water is two inches higher than the potatoes. Add about a teaspoon of salt to the water. Place the pot on the stove over high heat, covered.
When the water comes to a boil, lower the heat to a simmer. Cook the potatoes, covered, for 20-25 minutes. Check on any smaller potatoes sooner to see if they are done and need to be removed from the pot with a slotted spoon. The potatoes are done when you can easily insert a fork into the center.
While the potatoes are cooking, cook the green beans. Cook frozen green beans or bagged ones in the microwave according to the package instructions. (If using fresh green beans, rinse them and snap off the ends. Boil or steam them until they are cooked to your liking.) Drain and toss them with one teaspoon of olive oil and set them aside.
Chop up a peeled red onion until you have ⅓ cup. Set it aside.
Carefully remove the cooked potatoes with a slotted spoon and place them in a single layer on a large platter or sheet pan to cool.
When the potatoes are cool enough to handle but still warm, peel them with a small paring knife and discard the skins. Cut the potatoes in half, then cut those pieces in half. If needed, continue cutting the pieces in half until you have chunks that are about 1 ½ inches long.
Place the potatoes in a large serving bowl. Whisk the dressing ingredients or just add each one to the bowl separately. Use two spoons to gently toss the potatoes with the dressing to mix.
Add the red onions and green beans to the bowl. Stack a couple of basil leaves at a time and roll them up into a cylinder. Slice them up into ribbons and add them to the bowl. Gently toss everything together to mix. Garnish with additional basil, if desired.
Serve the potato salad at room temperature. Refrigerate leftovers for up to four days.
Green beans are traditional in this dish from Naples, but the potato salad still will be delicious without them. If you omit the string beans, be sure to scale back the amount of dressing you make.
You can boil the potatoes a day ahead. After cooking them, drain and pat them dry, then cover and store them in the refrigerator. You can leave the boiled potatoes whole or peel and cut them into chunks before refrigerating them.