Sanhedrin 42a says that the practice in E”Y then to say Baruch Mehadesh Hodashim is proper for women (Maharsha says women don’t remember the long braha, and the Einei Shmuel says women don’t bless with HaShem’s name.
The Gemara’s implication is that women ought to say something, if even the shorter version. This is Meiri’s peshat in the Gemara, and Hakham Ovadia Yosef paskens like this le ma’aseh (Yabia Omer OC 5:36).
Teshuvot Ktav Sofer (OC #34) says that women make some kind of braha, but not the one men say.
The Salmat Haim gives the reason “kol kevuda bat melech penima” (1:259). Others say that women are patur because Hava caused the pegam halevana (Shelah Sha’ar haOsiyot Kuf).
Maharam Shik (OC 90) says that when to say kiddush levana is based upon figuring out the calendar, and this was a skill performed only by men.
In a similar fashion, the Yeshu’ot Ya’akov (OC 426:1) writes that the beracha of kiddush levana is related to “haskalah and Torah She ba’al peh,” things in which women are not involved.
The overwhelming score of Ashkenazic poskim say a woman is patur.
I wonder what he’d say about whether a woman is obligated in birkat ha hammah.
Here is the Shelah inside:
The reason goes back to two ideas that have been linked: one being the legend that explains why the moon is smaller than the sun, and the other being the discussion of the consequences for women of Eve’s sin.
In the future the defect of the moon will pass away, and the
light of the moon will be like the light of the sun. . . . . It appears to me that for this reason women distance themselves from this commandment, even though they fulfill many positive time-bound commandments, such as shofar-blowing and [taking] the lulav [on Sukkot]. We have never seen women performing Kiddush Levanah, even if they are careful with all the prayers, because the first woman caused the defect of the moon, that is the sin of Eve, and they distance themselves because of the shame (bushah), even though a tikkun (repair) was found for them after this, in that they did not sin in the matter of the (golden) calf, and they did not listen to the first snake, who is the Satan, who is the Evil Inclination,
and therefore Rosh Hodesh was given to women that they should keep it more than the men. In any case the woman is the first cause (sibah rishonah) by which the seducer entered the world, and afterwards the men strengthened him with the (sin of the) calf, and we are still not purified.”