Philippine President Rodrigo Duterte’s approval ratings fell sharply in September to their lowest level since he took office nearly 16 months ago, following a series of political scandals and an outpouring of opposition to his flagship antinarcotics campaign.

The president’s net satisfaction rating fell 18 points to 48, classified as “good,” in a survey conducted late last month by independent pollster Social Weather Stations, compared with figures from its June survey. Mr. Duterte’s net trust rating fell 15 points to 60,...