Using a food processor fitted with a coarse grating disk, grate the potato and onion; discard any large pieces. Transfer the mixture to a clean dish towel and squeeze out any excess liquid, then transfer to a large bowl. Stir in the flour, egg, 3/4 tsp. salt, 1/4 tsp. pepper and the baking powder.
Trim and quarter 2 medium zucchini (about 8 oz.). Using food processor fitted with coarse grating disk, grate zucchini. Transfer to medium bowl and toss with 1/2 tsp. salt. Let stand, 10 minutes; squeeze to drain. Add zucchini to potato-onion mixture; squeeze mixture dry again. Add 2 tbsp. chopped fresh basil to batter before frying latkes.
In a medium skillet, heat 1/4 inch oil over medium-high heat until hot but not smoking. Working in 5 to 6 batches, drop 2 tbsp. scoops of batter into the pan about 2 inches apart. Using a spatula, flatten the batter into disks. Cook, turning once, until browned and crisp, about 5 minutes. (Lower the heat if the latkes darken too quickly.) Drain on paper towels.