The obligation of Kiddush on Shabbat applies equally to men and women. On Friday night, both men and women are required by Torah law to recite or hear Kiddush.
The Misva of Kiddush is exceptional in this regard, as women are generally exempt from "Misvot Aseh She’ha’zman Gerama" – affirmative Biblical commands that apply only at certain times. Kiddush, of course, applies only on Shabbat, and we might have therefore assumed that women are exempt from this obligation. The reason why women are included in the Misva of Kiddush stems from the Talmud’s comment that the two basic commands of Shabbat – "Zachor" ("Remember") and "Shamor" ("Guard") – were proclaimed together. "Zachor" refers to declaring the sanctity of Shabbat (Kiddush), while "Shamor" commands us to observe the Shabbat prohibitions. Since "Zachor" and "Shamor" were stated together, the Talmud comments, anybody included in the command of "Shamor" is likewise included in the command of "Zachor." Therefore, since women are bound by the command of "Shamor," the prohibitions of Melacha (forbidden activity) on Shabbat, they are similarly included in the obligation of Kiddush, which is derived from "Zachor."
Since men and women are equally bound by the obligation of Kiddush, a woman may, strictly speaking, recite Kiddush on behalf of a man. Of course, it is not customary for women to recite Kiddush on behalf of men, but if a man, for whatever reason, cannot recite Kiddush, such as in a case of illness, Heaven forbid, a woman may recite Kiddush for him. This applies even if the woman had already recited or heard Kiddush, and has thus fulfilled her obligation; she may still recite Kiddush on behalf of others, including men. The basis for this Halacha is the famous concept of "Arbut." The principle of "Kol Yisrael Arebim Ze La’ze," or "All Israel are responsible for one another," establishes that even after a Jew has fulfilled a Misva, he has not completely satisfied his obligation so long as there are Jews who have not fulfilled that Misva. Therefore, for example, a person who already heard the Shofar on Rosh Hashanah can nevertheless blow the Shofar for somebody else who has not heard the Shofar. Even if a person has blown the Shofar a hundred times on Rosh Hashanah, he may still blow the Shofar again for somebody else, who fulfills his obligation through that blowing. The same is true of Megilat Esther; a person can read the Megila dozens on times on Purim on behalf of others. Even though he has already fulfilled the Misva, his fulfillment is incomplete until all other Jews have also fulfilled the Misva. He is therefore qualified to read the Megila for others, no matter how many times he has already read it.
This applies to Kiddush, as well: a person can recite Kiddush on behalf of another person even though he has already fulfilled his obligation, because of the concept of Arbut. Hacham Ovadia Yosef rules that Arbut includes both men and women, and therefore a woman may recite Kiddush for a man even though she has already fulfilled her obligation. Thus, for example, if a husband is sick and incapable of reciting Kiddush, and his wife recited Kiddush herself while he slept, she may recite Kiddush again for him when he awakens.
By the same token, a person who recited Arbit on Friday night may recite Kiddush on behalf of somebody who had not recited Arbit. In the Arbit recitation, we recite the Beracha of "Mekadesh Ha’Shabbat," which fulfills the Torah obligation of Kiddush. The additional requirement to recite Kiddush over a cup of wine applies only on the level of Rabbinic enactment. Therefore, a person who recited Arbit has a lower level of obligation to recite Kiddush than somebody who has not recited Arbit. Nevertheless, one who recited Arbit can recite Kiddush on behalf of somebody who had not recited Arbit. Just as a person who recited Kiddush and fulfilled his obligation can still recite Kiddush on behalf of somebody else, similarly, and all the more so, a person who prayed Arbit and now bears only the Rabbinic obligation of Kiddush may recite Kiddush on behalf of somebody who bears a Torah obligation.
As mentioned, it is not customary for women to recite Kiddush on behalf of men. Nevertheless, this Halacha underscores the fact that women are no less obligated in Kiddush than men. Some women mistakenly approach Kiddush as a "man’s Misva" and do not make a point of listening attentively while the husband recites Kiddush. Both men and women are equally bound by this Misva, and both must ensure to fulfill this obligation properly.
Summary: The Misva of Kiddush on Shabbat applies equally to men and women. Strictly speaking, a woman may recite Kiddush for a man, though this is not customarily done. It is generally the husband who recites Kiddush, but the wife must ensure to listen attentively to each word.
