Are you asking for a percentage? Or an answer like "a lot" or "a little?" I don't like the way you've framed the question, but the answer is: "a lot." Myth surrounded the ancient Greek world; it pervaded life, informed it, complicated it. It was a basis for behaviors of all sorts, a reference point, a sort of moral code, and a legitimacy tool.
The leisure world of the Greeks was wholly dependent on mythology. The drama performances our 4th century Greek sees are mythology; so are the lyric poems, and the epic recitals. The architecture all around is decorated with mythic motif, and its interpretation depends on a broad knowledge of myth for understanding. Was it possible to go through life without knowing any myths? I suppose, but you'd have to work very hard at it. You were hearing myths, and internalizing them, when you were a toddler.
Mythology also infused and instructed ritual observance, which occupied a gigantic amount of mental territory in the average Greek person's life. Myths define and illustrate the roles of the gods and heroes; they demonstrate accepted and deviant standards of behavior in everyday life; they mirror and critique human behavior in archetypal, fable form; and they explain and canonize ritual behavior. Why am I sacrificing this fawn to Artemis? Why I am drowning this piglet in the Saronic Gulf in honor of Demeter and Persephone? I'm doing it no matter what, because ritual participation is key to my existence as an ancient Greek individual; but why? The answers are tied up in the mythology surrounding those gods, and those corresponding festivals.
There were many myths. Modernity probably doesn't know even half of all the "Greek myths" that existed in antiquity, and some were obviously more prevalent than others. Everybody knew about Zeus' many extra-marital exploits, with their fantastical metamorphoses, but we can probably assume that not everybody knew the particulars of the contest between Athena and Poseidon for who would hold sway in Athens (the fact that Athena gave an olive tree and Poseidon gave a salty spring). Many myths were local like this, and in their local environments they played powerful and important roles which were not as relevant on the other side of the sea. For instance: many local myths legitimized oligarchic claims to power, or reinforced leading families' claims to territory or political authority. Aristocratic families in Attica claimed moldy old Mycenaean tholos tombs as burials of their heroic ancestors, and offered cult to those imagined ancestors. They certainly had myths to tell the stories of those forgotten heroes, which they no doubt doggedly repeated to their fellow citizens. Likewise, individual poleis had their own local tales of common gods, like what Artemis got up to at Ephesus, or what Apollo did at Delphi, or what Theseus did at Athens. The identities of many poleis were intimately tied up in ther mythic founding event, or the like.
We often get, in Greek literature, a very "meta" look at how myth worked. When Achilles gets his sex slave taken away from him and decides to sulk in his tent instead of helping to fight in the Trojan War, the men who come to convince him to shape up use a myth to show him the error of his ways (Meleager, another hero who had refused to fight because he was pouting). The myth illustrates life and provides a sort of framework of consequences to behavior. In this way, the iconic nature of Greek myth (with human-like deity) is useful and adaptable.
So no, there weren't very many Greeks who didn't know the myths. No one knew them all, but most almost certainly knew most in one way or another. They weren't just stories to memorize (although that was part of Greek education to be sure), but rather a sort of basic fabric upon which Greek life was structured.
As always, a fairly approachable source for Greek religious questions is Bremmer's Greek Religion (1994).
