The morning after the election of Donald J. Trump, Kirsten Gillibrand woke up and began crying. “Bawling,” she corrected herself.

After spending the next two months “deeply depressed,” the junior senator from New York experienced what she called the most inspiring day of her political life: the women’s march. And almost two years to the day after that, Ms. Gillibrand sat in a Manhattan wine bar, holding a glass of pinot noir, and described why she believes the country and the Democratic Party need an unabashedly feminist campaign for president — and why she thinks she’s the candidate to run it.

“You understand, this is my space,” Ms. Gillibrand said, in an interview. “I don’t know if my party will get as far as I will go on a lot of these issues. But I believe in them so strongly.”

At a moment when women are ascendant in Democratic politics and polls show a record gender gap between the parties, Ms. Gillibrand and her advisers see an opportunity to ride a wave of women’s political energy right into the White House. While multiple women are serious contenders for president for the first time in American history, Ms. Gillibrand is the only one who is making running as a woman, for women, the central theme of her candidacy.

Whether her campaign is successful or not, Ms. Gillibrand’s approach will test how much views have changed on issues like discrimination, sexual harassment and female leadership, after two years of an administration that has embroiled the country in an emotional debate over gender bias.