There is a mitzvat aseh (positive commandment) from the Torah to hear the shofar (ram's horn) on Rosh Hashanah (Jewish New Year). This, however, is known as a mitzvat aseh shehazman geramah. It is a time-bound positive commandment. Therefore, women are exempt from time-bound positive commandments, so women do not have an obligation from the Torah to hear the shofar. Nevertheless, women are encouraged to come and perform the mitzvah (commandment), and they get sachar, they get reward for doing the mitzvah. This is very common in our community and in many communities around: women come and flock to the Beit Knesset (synagogue) to hear the shofar and receive reward for the mitzvah.
However, they don't get the same reward as men because they're not commanded to do so. The Yetzer Hara (evil inclination) is stronger when people are commanded, to get them not to do it. Therefore, the reward for men is greater. However, women should be encouraged to come, but not at the expense, A, of a man. A woman should not say, “I want to go hear the shofar, so my husband has to stay home and watch the kids, and I should get the mitzvah and he shouldn't.” Nor should they alternate days, because there's a mitzvah to do it on both days, with the husband going on day one and the wife going on day two, and the husband not going. So it should not come at the husband's expense, because the husband has the obligation and the wife does not have the obligation.
It also should not come at the expense of bringing very small children to the Beit Knesset who are going to cause a ruckus and end up not allowing people to properly hear the shofar and fulfill the mitzvah. If the children are older and the mitzvat chinuch (educational obligation) applies to them, then they should be brought in order to train them. But if they're very little and there's no chinuch (education), and they're coming and causing problems for people trying to hear the shofar and fulfill their mitzvah, then a woman should stay home and take care of her kids. She should not cause an expense for everybody else when she is not obligated in the mitzvah. Have a great day.