Yes, they do. According to the South American Explorer’s Club (a very useful site for anyone going anywhere in South America, but especially off the beaten path):
“The official currency of Ecuador is the US dollar. Dollar bills circulate in denominations of $1, $2, $5, $10, $20, $50, $100. Coins are used alongside the equivalent size and value Ecuadorean coins for $0.01, $0.05, $0.10, $0.25, $0.50, and $1. Be sure when you leave Ecuador, you do not have any Ecuadorean coins as they have no value outside the country. Many merchants are reluctant to take bills over $20 because of the potential of counterfeit bills as well as a lack of change.”
They prefer American Express Traveler’s Checks, and most merchants accept Visa, MasterCard, Diners Club, and American Express.