CARACAS—Venezuela is run by an anti-American, socialist government. But the country increasingly relies on a potent symbol of American capitalism: The U.S. dollar.

The trend is sparking a growing inequity between Venezuelans with dollars and those stuck with an increasingly worthless local currency, the bolivar. The divide was clear during a recent national blackout. With no debit card machines working, and few people carrying the vast quantities of bolivars needed to buy goods, the greenback became the primary means of payment....