Place the potatoes in a large pot of salted water and bring to boil. Cook until just fork tender, about 15 minutes. Fish out the potatoes with a spider or slotted spoon and place them in a bowl of ice cold water to stop them from cooking.
Shuck the corn and break each ear in half. Cook in the same boiling water for 5 to 7 minutes until tender but not soft. Remove the cooled potatoes to a dish-cloth to drain. Immerse corn in the same ice bath until cool. Cut each potato into quarters and place in a large bowl.
Remove corn from water and also let drain. Use a chef's knife to cut the kernels off each ear. Add kernels to bowl. Add grape tomatoes, onion, and whole basil leaves. Add olive oil and lemon juice and toss gently to combine. Season, to taste, with salt and pepper. Serve immediately.