The Trojan Women, a tragedy written by Euripides and first performed in 415 BCE, and Lysistrata, a comedy written by Aristophanes and first performed in 411 BCE, were both written during the Peloponnesian War. It was a period of strife between Athens and Sparta from 431–421 BCE , and renewed in 415-404 BCE. The second period involved many more city-states in and around the Peloponnesian peninsula.
The two periods of war were separated by a period called the "Peace of Nicias," which was a peace in name only. The largest land battle fought during the Peloponnesian War occurred in 418 BCE, during the so-called "peace" period.
Trojan Women and Lysistrata were both written during the "Second War" period. Women are protagonists in both plays—Hecuba in The Trojan Women, and Lysistrata in Lysistrata.
Beyond that, the two plays have little in common, and there's little to support the idea that Lysistrata is a response to The Trojan Women.
Euripides is mentioned in Lysistrata, but so, too, is Aeschylus. There's no mention of The Trojan Women in Lysistrata, and the reference to Euripides in Lysistrata hardly takes Euripides or The Trojan Women to task.
CHORUS OF MEN. Euripides was right! Women are shameless!
No doubt Aristophanes knew about The Trojan Women. He might even have seen it performed at the Festival of Dionysus in Athens. Lysistrata was possibly performed four years later, or at a lesser festival, the Lenaia.
The Trojan Women is an allegory, and it stands today as one of the first true anti-war plays. It functions as a scathing commentary on the brutal subjugation of the island of Melos by the Athenians. The Athenian army destroyed Melos because the people of Melos chose to remain neutral in the Peloponnesian War.
The Trojan Women isn't a feminist play, and neither is Lysistrata. Many productions of Lysistrata approach the play from that perspective.
The underlying tone of Lysistrata, as well as Aristophanes and Greek society's view of women, is clearly established in the opening scene of the play.
LYSISTRATA. Calonice, there are many things about us women sadden me, considering how men see us as shrews and cunning rogues...
CALONICE. As indeed we are!
At the end of the scene, when the women of Athens and other city-states agree to go on a sex strike until the men end the war, Lysistrata remarks:
LYSISTRATA: By Aphrodite, that’s the spirit. We’ll show them that women are a strong and wily sex.
Lysistrata herself is the exception to the Greek perception of women, and she's portrayed as such. When the men accede to the women's demands and end the war, the women resume their "womanly" roles in Greek society.
Lysistrata isn't an anti-war play. The morality or viability of the Peloponnesian War isn't discussed in the play. It's the length of the war and the time that the men are away from their wives and children that Lysistrata and the women are protesting.
LYSISTRATA. Are you not sad your children's fathers
Go endlessly off soldiering afar
In this plodding war? I am willing to wager
There's not one here whose husband is at home.
The women aren't concerned with the men's reasons for initiating and sustaining the war. The sex strike is simply one strategy, devised by a woman, for ending it.