The intelligentsia were definitely suspicious of myths. Herodotus begins his history with a rationalization of stories like the Rape of Europa. Livy begins his by the caveat that the stories about the history of Romans before the founding of the city are more appropriate for poetry than history. Even Ovid's Metamorphoses, the text that transmits most these stories, is so tongue-in-cheek it's difficult to take it as meaning to be a serious treatment.

Among the philosophers, the blatant immorality of the Gods was a real problem for ideas about divinity, and people like Euhemerus produced arguments that the stories are embellishments of stories about real people, not gods at all.

Clearly these sorts of people are not the masses, but its really difficult to know what, say, a dishwasher thought; there's no lower class writing.

Some people were interested in collecting myths - the library of Apollodorus is a terribly dry encyclopedia of myth. However an interest in myth for antiquarian purposes doesn't imply the antiquarian believed any of it.