The Gemara in Masechet Megila (7) introduces the obligation to rejoice on Purim through drinking intoxicating beverages: "A person is obligated to rejoice on Purim to the point where he cannot distinguish between Haman and Mordechai." This passage is generally understood to mean that one should drink wine until he becomes a bit dizzy so that he cannot think properly as he normally does.
The question arises as to whether this obligation applies to women. Women are included in all the Misvot of Purim, such as hearing the reading of the Megila, giving charity to the poor, sending Mishlo'ah Manot, and so on. Seemingly, then, they are likewise included in the obligation to become inebriated on Purim.
However, Rabbi Efrayim Greenblat of Memphis (a student of Rabbi Moshe Feinstein), in his work Rivevot Efrayim (1:458), rules that women should not, in fact, drink intoxicating beverages on Purim. He bases this ruling upon a number of arguments. Firstly, he explains the requirement to drink on Purim as a celebration we conduct in anticipation of the ultimate destruction of the descendants of Amalek. We drink not merely in celebration of the downfall of Haman, a member of Amalek who lived centuries ago, but also in eager anticipation of the day in the Messianic era when nothing will remain of Amalek, and there will be no more hatred towards the Jewish people. Accordingly, Rav Greenblat contends, since women are not included in the obligation to wage war against the nation of Amalek, they are likewise not included in the obligation to drink on Purim. This obligation relates to the ultimate destruction of Amalek, and thus it applies only to men, who bear the responsibility to wage this battle.
Secondly, Rav Greenblat draws a comparison between drinking on Purim and the obligation of candle lighting on Hanukah. The Hatam Sofer (Rabbi Moshe Sofer of Pressburg, Hungary, 1762-1839) ruled that even according to the Ashkenazic custom that each member of the household personally lights Hanukah candles, women fulfill their obligation through their husband's lighting. He explains that in Talmudic times, the Hanukah candles were lit outdoors, near the public domain, and it would be immodest for women to go to the public area to light Hanukah candles. Similarly, Rav Greenblat claims, it would be immodest for a woman to drink to the point of inebriation, and thus the obligation to drink on Purim does not apply to women.
Rav Greenblat points to a number of sources indicating that it is deemed inappropriate for a woman to drink intoxicating beverages. The Meiri (Talmudic commentator, France, 1249-1315), for example, makes such a comment in reference to the law of "Ben Sorer U'moreh" – the wayward son who is put to death due to his gluttonous behavior, ingesting excessive amounts of meat and wine. The Torah speaks only of a son, and not of daughters, because, as the Meiri explains, it is uncommon for a girl to be drawn after such gluttonous excesses. Rav Greenblat infers from this comment that it is not natural or proper for a woman to indulge in intoxicating beverages.
Another source is a comment the Sha'ar Ha'siyun (notes appended to the Mishna Berura commentary to the Shulhan Aruch), in the laws of Zimun (199:6). The Sha'ar Ha'siyun claims that women do not count towards a Zimun (quorum of three men required for the special Beracha before Birkat Ha'mazon) because the Zimun was traditionally conducted over a cup of wine, and it is unbecoming for a woman to drink wine. Rabbi Greenblat notes that if women were excluded from Zimun because it customarily involved drinking a single cup of wine, then certainly it would be inappropriate for a woman to drink and become inebriated, even as part of the Purim festivities.
We might also add that the Gemara presents the obligation to drink on Purim with the phrase, "Hayav Inash…" – "A man is obligated" – clearly suggesting that this obligation applies only to men.
We should note that although women are obligated to drink the four cups of wine at the Seder on Pesah, that drinking is not intended for the purpose of intoxication. Indeed, this Misva can be fulfilled by drinking grape juice. On Purim, however, the drinking is clearly done with the aim of becoming inebriated, which is inappropriate for women. It is thus customary for women not to drink alcoholic beverages on Purim for the purpose of getting drunk.
Summary: Although Halacha requires drinking intoxicating beverages on Purim, this obligation does not apply to women, and it is in fact inappropriate for a woman to drink intoxicating beverages, even on Purim.