There are several definitions of "exploit," most of them having good denotations and good connotations. I'm assuming though that you mean "exploit" in its negative meaning since you couple it with "gender stereotypes."
- exploit: to use selfishly for one's own ends. To make use of selfishly or unethically. [T]o take advantage of, esp unethically or unjustly for one's own ends. (Random House, American Heritage, and Collins Dictionaries)
- stereotype: A conventional, formulaic, and oversimplified conception, opinion, or image. Sociology--a simplified and standardized conception or image (American Heritage and Random House Dictionaries)
"Stereotype" has negative denotation and connotation. It was originated, therefore unknown until, between 1790 and 1800.
Consequently, to suggest that Aristophanes exploited gender stereotypes in any way is without foundation. In ancient Greece, people were seen as having specific functions to perform. To elaborate, Robin Lane Fox, in Pagans and Christians, makes it clear that gender roles in ancient Greece were far greater than stereotypically believed. He presents clear evidence that women had significant power as benefactresses and city planners.
It is a universal fallacy that because works like the comedies of Aristophanes discuss certain social or ethical problems, they are inspired by them. (Jack Lindsay, Forward, Lysistrata)
The question you are really asking is: What gender stereotypes can we recognize from our own cultural orientation as being present in Aristophanes Lysistrata, recognizing that Aristophanes' perspective would not have concurred with ours and that we don't even yet have an adequate picture of ancient Greek society?
In the opening of the drama, we might perceive a gender stereotype in the very premise of the play: the men have been at war for twenty years and the women at home keeping the home fires burning (and the city running). The conflict of the play, which is embedded in a plot devised by Lysistrata, the main character, may also be perceived as a stereotype: the women choose not to do what the men demand of them to do causing a social battle divided along gender lines of he against she. Lysistrata's plan itself might be seen as representing gender stereotypes although the plan is meant to sabotage and reverse the gender stereotype: women intend to withhold sexual favors. This anti-gender stereotype plan has a specific cultural objective: sexual activity will be withheld until the ongoing twenty year war is brought to a permanent end.
Another example of what might be perceived as gender stereotypes is the occupations the women protest a need to return to with some urgency. One must go home to tend her "Melisian wool" to save it from moths; another, her unstripped flax, and she must "flay it properly"; another has an urgent pregnancy that she is on the verge of delivering (it is miraculous in that it sprang up in one day and turns out to be a helmet in disguise).
I must get home. I've some Milesian wool
Packed wasting away, and moths are pushing through it.
Fine moths indeed, I know. Get back within.
By the Goddesses, I'll return instantly.
I only want to stretch it on my bed.
You shall stretch nothing and go nowhere either.
Must I never use my wool then?
If needs be.
How unfortunate I am! O my poor flax!
It's left at home unstript.
So here's another
That wishes to go home and strip her flax.
I'll drop it any minute.
Yesterday you weren't with child.
But I am today.
O let me find a midwife, Lysistrata.
it's Athene's sacred helm,
And you said you were with child.
And so I am.
Then why the helm?
[As] a laying-nest in which to drop the child.
More pretexts! You can't hide your clear intent