Ginsburg, who died on Friday at age 87, was born in Brooklyn in 1933 and raised in the borough.
The campaign to rename the building after the "proud daughter of Brooklyn" began two years ago, said Brooklyn Borough President Eric Adams.
"I take heart in knowing that young girls and boys who pass by the Ruth Bader Ginsburg Brooklyn Municipal Building will know her name, learn from her example, and pick up the baton to run their own relay toward a more just, equitable and fair America," Adams said in a statement Tuesday.
"As a lawyer, jurist, and professor, she redefined gender equity and civil rights and ensured America lived up to her founding ideals -- she was a monumental figure of equality, and we can all agree that she deserves a monument in her honor," Cuomo said in a statement.
"She persevered despite several bouts of cancer and was present every single day to participate in the strengthening and safeguarding of our democracy," Cuomo continued.
"We remember proudly that she started her incredible journey right here in Brooklyn," Cuomo said. "Her legacy will live on in the progress she created for our society, and this statue will serve as a physical reminder of her many contributions to the America we know today and as an inspiration for those who will continue to build on her immense body of work for generations to come."
Cuomo said in the coming days he'll appoint a commission to select an artist and provide recommendations on possible locations.
Landmarks across New York state were lit blue on Saturday for Ginsburg.
"Blue is the color of justice and was reportedly Justice Ginsburg's favorite color," Cuomo's statement said.
The landmarks include Niagara Falls, One World Trade Center, Grand Central Terminal and the Kosciuszko Bridge in Queens.