Essentially, do we know how much of an influence myths were in the life of a regular Greek or Roman citizen? Were they similar to how religion is today, ie. were there people who actively worshipped the deities?
I can't speak to Roman religion.
As for ancient Greek religion, myth played an important role in cult practice. Despite Pan-Hellenic similarities, local cult had differing and even conflicting myths about the origins and functions of the gods. Furthermore, “no two Greek communities worshiped the same set of gods; individual gods had very different prominence and roles in different places.” (1) This goes into the epithet system. In example, If one needed the aid of Artemis Soteira, Artemis Eileithyia (of childbirth), and Artemis Orithia (of the dawn), then three separate prayers and offerings would be made. One could pray to Artemis Soteira to be spared losing a battle but not to ease childbirth, even though that was an area she presided over as Artemis Eileithyia.
The discrepancy of differing myths didn't seem to be a concern because there would be shrines to the same god under different epithet in the same city.
One way that we can say life was directly impacted by myth was during festivals. Myths would often be religiously acted out. During the Athenian Anthesteria, maidens would swing on wooden swings in order to honor Erigone, who hanged herself after the murder of her father. However, there's a different Erigone myth that makes her the sister of Orestes, who killed their parents, and Orestes was honored within the same festival. Both versions of the myth were represented during the three day festival.
I think the celebration of festivals, the numerous offerings and prayers that we know about, the shrines and temples, etc. shows that the ancient Greeks believed in their gods/had what we would consider faith.
I hope this helps!
General references:
Burkert, Walter, Greek Religion. Cambridge, MA: Harvard University Press, 1985.
It you're interested in learning about the religious system, I would suggest Ancient Greek Religion by Jon D. Mikalson and Ancient Greek Cults by Jennifer Larson. Burkert's work is great and informs the two I've recommended, but its dense. I found it hard to get through when I first started studying this.