Euripides' Medea was first staged in 431 BCE in Athens. This year was an important one because it marked the beginning of the Peloponnesian War between Athens and her allies and Sparta and her allies. Euripides composed his play as this war was breaking out. Tensions between Athens and Sparta had been running high for many years. In 446/445 BCE, the two city-states had entered into a 30-year peace treaty, but in the final years of the 430s that treaty began to crumble.
The breaking of this treaty is important with respect to Euripides' play, which has a lot to say about the breaking of oaths, namely the marriage vows that Medea believes binds her and Jason. With respect to the Peloponnesian War, Sparta regarded Athens as having violated the treaty when, in 432 BCE (the year before Euripides' play) they became involved in a military conflict involving the Corinthian colony of Potidaea (the Spartans were allies of the Corinthians). Interestingly, Corinth is the setting for Euripides' play. Thus, Euripides gives his fellow Athenians something to think about when he brings to their ears Medea's curses against oath-breaking Jason.
As for Medea,
that poor lady, in her disgrace, cries out,
repeating his oaths, recalling the great trust
in that right hand with which he pledged his love.
(Ian Johnston translation)
Also, Jason's silver-tongued defense of himself later in the play may have reminded some in Euripides' audience of the clever orators within their own city. Jason attributes his success to Aphrodite, not Medea, and claims that he did her a favor by bringing her from a land of barbarians to Greece.
In the examples cited above, we encounter what modern students of the play will regard as subtle hints at Athenian culture.
The most obvious example of Athenianism, however, occurs when King Aegeus, the mythical king of Athens, appears and offers Medea asylum in Athens in exchange for her help in having children. We should note again the presence of an oath. Aegeus swears that he will assist Medea and mythical tradition tells us that Aegeus did eventually take in Medea.
What Euripides' Athenian audience probably knew, but what Aegeus did not know at the time he swore his oath to Medea, was that Medea was going to kill her children before she left Corinth. Thus, Aegeus, the mythical king of Athens, was going to grant asylum to a child-killer. Moreover, mythical tradition held that Aegeus would marry Medea and that she came very close to getting Aegeus to kill his own son Theseus, the mythical founder of democracy.
Thus, when trying to search for the concept of Athenianism in Euripides' Medea, we should look especially at the words and deeds of Aegeus, but also what the play has to say about oaths. Ideally, a "good" Athenian would not break his oath. The people of Sparta and Corinth, however, would certainly have a different view of that virtue in 431 BCE.