Put the potatoes in 2 large pots and cover with water. Add salt and bring to a boil. Reduce the heat and simmer until the potatoes are tender, about 40 minutes. Drain the potatoes and let cool slightly.
Peel the potatoes. Halve them lengthwise, then slice crosswise 1/3 inch thick. Transfer the warm potatoes to a large bowl. Add the olive oil, vinegar, salt and pepper and toss gently to coat the potatoes.
Shortly before serving, add the crumbled goat cheese, garlic, oregano and basil to the potato salad and toss well. Transfer to a platter and serve.
The potato salad can be prepared through Step 2 and refrigerated overnight. Bring to room temperature before serving.