Put the potatoes in a pot with cold water and add some salt. Cover, bring to a boil and boil until fork-tender, 10 to 15 minutes. Drain, cut in half and transfer to a bowl.
Toast the pine nuts in a skillet over low heat until just light golden, 3 to 4 minutes.
Mix together the mayonnaise, olive oil, pesto, lemon juice and salt and pepper to taste in a bowl until creamy and smooth. Taste and adjust the seasoning.
Pour half the dressing over the potatoes and toss to combine. Sprinkle over the pine nuts and stir in. Add more dressing to taste.
Cover and refrigerate the potato salad for a few hours. To serve, sprinkle lots of small basil leaves over the top.