GUVECCI, Turkey—A little past midnight in this dusty border town, the only sounds other than crickets were smugglers herding newly arrived refugees into minibuses bound for a nearby city.

A slender woman from the Sahel al-Ghab region in northwestern Syria, where fighting between rebels and government forces has raged in recent weeks, is among the first of several waves of exhausted refugees to reach Guvecci on this September night. She said she and her husband paid a smuggler $130 to be led on a 15-minute dash by foot across...