WASHINGTON—Sen. Kirsten Gillibrand (D., N.Y.) formally announced her bid for the U.S. presidency on Sunday, releasing a campaign video and tweeting her decision to join the list of candidates in the crowded Democratic primary.

“I’m running for president,” Ms. Gillibrand announced on Twitter. “Let’s prove that brave wins.”

Ms. Gillibrand, 52 years old, is pitching herself as an advocate for women and families, among other things. She is calling for affordable child care, universal pre-kindergarten, and a $15-an-hour minimum wage.

Ms. Gillibrand has been considered a candidate since January, when she announced the formation of an exploratory committee on CBS’s “The Late Show with Stephen Colbert,” and said she would run.

So far this year, she has already swung through Iowa and New Hampshire, home of the first caucuses and primaries in the nation.