Thinking about the idea of Helen's desire for kleos, and how she is limited in the ways in which she can achieve it as a woman, it is rather interesting to see how the most 'obvious' solution to it is avoided — through children.
The main consensus surrounding Helen's number of children is that she gives birth to one, a girl called Hermione. Some sources bring up several others, but the earliest and most consistent child mentioned (and birthed by Helen) is her:
He was bringing Alector's daughter from Sparta for his son, mighty Megapenthes, grown up, who'd been borne him by a slave. The gods no longer made a child appear for Helen after she gave birth to her first child, lovely Hermione, who had the form of golden Aphrodite. (Homer, Odyssey, OD.4.10 - ca. 700 BCE)
MENELAUS [Laurentian Scholiast on Sophocles' Electra, 539]: 'And she (Helen) bare to Menelaus, famous with the spear, Hermione and her youngest-born, Nicostratus, a scion of Ares.' (Hesiod, Fragments, CW.F70 - ca. 650 BCE )
Now Menelaus had by Helen a daughter Hermione and, according to some, a son Nicostratus; and by a female slave Pieris, an Aitolian, or, according to Acusilaus, by Tereis, he had a son Megapenthes; and by a nymph Cnossia, according to Eumelus, he had a son Xenodamus (Apollodorus, Library, 3.11.1 - ca. 100 CE)
When Orestes became king of the Lacedemonians, they themselves consented to accept him for they considered that the sons of the daughter of Tyndareus had a claim to the throne prior to that of Nicostratus and Megapenthes, who were sons of Menelaus by a slave woman. On the death of Orestes, there succeeded to the throne Tisamenus, the son of Orestes and of Hermione, the daughter of Menelaus (Pausanias, Description of Greece, 2.18.6 - ca. 174 CE)
It seems the later the sources get, the more children are added (look at that Apollodrus extract! Menelaus has been around!); but even then, Hermione is always mentioned, and often mentioned as the only legitimate child of Helen and Menelaus.*
*Yes the Scholiast on Euripides, Andr. 898 Cypria fragment does mention Pleisthenes the third born child, but... we're ignoring him (lol). As far as I'm aware he's only mentioned once and also no confirmation he's legitimate. Sorry Pleisthenes.
Now, let's talk about that a little. Helen giving birth to a daughter, and proceeding to not have any more children after that raises an important point — Helen does not produce an heir for her husband. Helen, as Menelaus' wife is expected to do that but she simply... doesn't. She has one child and that's it. Sometimes that one child isn't even intentional:
And she (Helen) bare neat-ankled Hermione in the palace, a child unlooked for. (Hesiod, Fragments, CW.F68 - ca. 650 BCE)
unlooked-for = unexpected; unforeseen.
However, as Ruby Blondell mentions in 'Helen of Troy: Beauty, Myth Devastation':
"In normal circumstances such failure to bear a son would jeopardize a Greek wife's position, since it denies her a woman's primary avenue to status within her husband's household. Since she did not relocate upon marriage, however, Helen needs no sons to establish her status at Sparta. Moreover, the absence of a son helps her maintain her independence. When a son grows up, he becomes another source of male authority over his mother." (pg. 32-33)
She has different circumstances from her peers; she doesn't need children to retain her status. In fact, if she were to bear a son it might be worse for her position.
This would explain her reluctance for children, if the lack of them was indeed a choice. Nonetheless, it's worth exploring the possibility that it is unintentional. There's certainly evidence for it being so, if we look at how Hermione fares in adulthood with her own fertility:
(to Andromache) but as for thee, slave and captive, thou wouldst fain oust me and secure this palace for thyself, and thanks to thy enchantment I am hated by my husband; thou it is that hast made my womb barren and cheated my hopes (Euripides, Andromache, 117 - ca. 425 BCE)
Helen is struggling to have children, and now Hermione is struggling to have children — like mother, like daughter. Is the infertility hereditary? Or perhaps there are some godly influences afoot:
TYNDAREUS Scholiast on Euripides' Orestes 249: Stesichorus says that while sacrificing to the gods Tyndareus forgot Aphrodite and that the goddess was angry and made his daughters twice and thrice wed and deserters of their husbands (Hesiod, Fragments, CW.F67 - ca. 650 BCE )
While this extract focuses only on Tyndareus' daughters and their infidelity, it wouldn't be a stretch to say that the curse of a god could extend to multiple generations of a family (e.g. the house of Atreus) and in other ways — in this case, not being able to get pregnant/have a successful pregnancy.
However, considering the infertility as an intervention by the gods, I don't think that this is necessarily a 'curse'. This feels more like fate. That, regardless of whether it is Helen's choice to have just Hermione or not, it is destined to happen. The subtext being 'just Hermione' and no more.
Blondell mentions that in Theocritus' poem The Epithamaly of Helen, the chorus pray that Helen bears Menelaus a child that resembles herself. If we take the quote from The Odyssey earlier, of Hermione 'who had the form of golden Aphrodite', we can assume that prayer is fulfilled. Blondell also says:
"There is something a little strange, however, about the chorus's prayer. ... since there is no doubt about a woman's identity as a mother, the crucial point, for a husband, is that his wife's offspring should resemble him. It seems strange too, at a wedding, to pray in effect for a daughter as opposed to a son and heir." (pg. 32)
Helen will have a child, and it will resemble her, but that's it. No sons, no heirs.
Enter stage left: Zeus' plan to reduce the population of demigods by having a beautiful daughter, who will instigate a long and bloody war.
How do you get such a daughter to instigate a war? By leaving her husband because she desires more than the cards she has been dealt.
If Helen does desire kleos, but is essentially barred from achieving it in the ways a woman traditionally does, then she must seek it elsewhere.
In choosing to leave with (effeminate, unconventional) Paris: If she doesn't wish to have children, maybe she thinks she can escape the 'inevitable' fate of all wives to bear a son and be known only as their mother — she will seek kleos elsewhere. If she does wish to have children, maybe she has tried to bear children, many, many times after Hermione and cannot, and in that shame of failing to meet the minimum standards of being a woman — she will seek kleos elsewhere.
Whatever her feelings are on the matter, fate remains the same: Helen will have a daughter that resembles her, and Helen will go to Troy with Paris. The facts still remain the same too: Helen cannot bear a son, and this separates her from womanhood. She cannot successfully be a woman by society's standards, and cannot make her legacy her children.
She must find kleos in other ways.
