Scrub 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 and cover. Cook the potatoes for 20-25 minutes, checking on any smaller ones to see if they are done sooner 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 your frozen green beans 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 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. Drizzle them with 2 tablespoons of oil and 5 tablespoons of vinegar. Add the red onion, salt, garlic powder and pepper. Gently toss the potatoes with two spoons to mix. Add the green beans to the bowl. Stack a couple 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 three days.