Although we have no clear idea about the opinions of the illiterate majority, the Roman elite was generally dismissive of the myths... It was almost universally agreed that the gods existed—even Epicureans admitted that—but it was almost as universally thought that the myths misrepresented the gods. By the second century CE, most Greek and Roman intellectuals either rejected the myths outright (the Epicurean approach), allegorized them into philosophical respectability (the Stoic approach), or claimed that the traditional gods were not gods at all, but mischievous daimones—spirits of the air—sent by the actual gods (the Platonic approach). This last interpretation became increasingly popular in Late Antiquity, when Neoplatonism emerged as the dominant philosophical tradition. By the time Constantine became Christian, very few educated Romans took their myths literally.
Christianity merely added another layer of allegory to the traditional pagan approaches to myth. Christian scholars resurrected the ancient allegorizing approach we call euhemerism (which claimed that the traditional gods were ancient human kings and heroes), and enthusiastically endorsed the idea that at least some myths were based on the deeds of demons masquerading as gods. The myths also offered conveniently low-hanging fruit for sermonizing; in the City of God, for example, Augustine describes Romulus' murder of Remus as characteristic of the sinful earthly city.
Garret Ryan, addressing the question "The Roman founding myth seems to be directly related to their pagan religion. Once the Empire was predominantly Christian, how did they come to terms with their pagan origins? Did they change their founding myth?" on /r/AskHistorians