This question is essentially communal, and its ramifications for the community’s identity, cohesion, and avodat Hashem play a major part in any decision. Arriving at a balanced decision entails consultation with a local religious authority who is in position to judge what will be of most spiritual benefit to the specific community. In cases like these, communal customs and norms can have greater halachic weight than written sources. Our response is intended as a preliminary discussion of the general issues.
In our article, "Contact with a Sefer Torah," we discuss the halachic sources permitting a woman to hold a sefer Torah. Once we take that as a given, the question becomes more specific. Jews are duty bound to show a sefer Torah honor when it passes by (Yoreh De’a 282), though one can debate whether deliberately trotting the Torah scroll around the synagogue is the greatest show of respect to it (Yalkut Yosef Keri’at Ha-Torah 134: Hotza’at Ve-hagbahat Ha-Torah 10). Given that most communities try to enable as many male congregants as possible to come close to the Torah during the processional, there is no strict halachic reason to deny women the opportunity to come close to the Torah. This sort of closeness to the Torah can be spiritually significant.
In some synagogues, the male carrier of the Torah deliberately passes next to the mechitza or the entrance of the women’s section so that woman who desire to touch and kiss the Torah have the opportunity to do so.
Ultimately, as stated above, whether the Torah should be carried through the women’s section (and how to manage it) is a communal decision that depends on the nature, character, and customs of the community.
If a community does reach a decision to pass the Torah through the women’s section, it is preferable for reasons of modesty for a woman to carry the Torah through a woman’s section rather than have a man march through it. Also, the logistics of passing the Torah should be treated with sensitivity (one possibility might be to place the Torah down first), so as to minimize mixing between genders in synagogue and to avoid touching.
See more Q&A here.