In order for the audience to sympathize with Medea, there first has to be a basis for that sympathy within Medea herself. In other words, the audience must be shown, or otherwise assume that, deep down, Medea is essentially a good and noble person.
At the beginning of Euripides's Medea, the Nurse provides the audience with the background of the play. Medea and Jason have fled to Corinth with their two sons after Medea took revenge on King Pelias, who tasked Jason with obtaining the Golden Fleece. Medea caused Pelias to be murdered by his own daughters. She also murdered her own brother, Apsyrtus, in order to aid in their escape to Corinth.
Jason has since forsaken Medea and married Glauce, the daughter of the local king, Creon. Medea is overcome with grief and rage, and vows revenge against Jason.
The Nurse, who loves Medea, nevertheless knows all too well that Medea will wreak a terrible vengeance on Jason. The Nurse also fears that Medea and Jason's young sons will play a part in Medea's revenge against Jason.
An Attendant appears with the children and tells the Nurse that Creon intends to banish Medea for vowing revenge against Jason. This causes the Nurse to be even more concerned for the safety of the two boys.
The Nurse confides her fears to the Attendant.
NURSE. Go: run into the house, my little ones:
All will end happily! ... Keep them apart:
Let not their mother meet them while her heart
Is darkened. Yester night I saw a flame
Stand in her eye, as though she hated them,
And would I know not what. For sure her wrath
Will never turn nor slumber, till she hath ...
At this point, Medea's voice is heard from inside her home calling out in grief and pain, and the Nurse and the Attendant break off their conversation.
In time, Medea affects her revenge. She kills her children, Creon, and Creon's daughter, Glauce, leaving Jason alive to suffer the rest of his life in loneliness and grief.
Although Medea suffers from excessive selfishness and pride (hubris)—flaws shared by many of the tragic characters in ancient Greek tragedies—her greatest and most tragic flaw (her hamartia) is an excess of passion. Medea is driven to her murderous behavior by her excessive jealousy, hatred, rage, and overwhelming need to take revenge on anyone who wrongs her.
It's true that Medea is wronged by Jason and Creon, and she's suffered terrible misfortune in her life, but is Medea a person for whom the audience can feel sympathy? Is Medea any less to blame for her circumstances than Pelias, Creon, or Jason?
While many audience members can feel pity for Medea and experience fear at her self-destruction—which Euripides no doubt intended them to feel—few members of the audience can truly sympathize with Medea or emphasize with her by putting themselves in Medea's situation. They simply wouldn't kill their own children, no matter the motivation or provocation.