The lawmakers were joined by 500 community members, including parents and caregivers at the launch of the "Strong Families, Strong New York" campaign.
"There are times in life when family comes first -- like when a child is born, a loved one is sick, or a parent is dying -- and I believe everyone deserves the right to be there in those times," Cuomo said at the event, located in a YMCA building.
"If you have a family member is is passing away, you should be there," he continued. "I know the feeling. I went through it last year with my father. That is the time you need to say things that need to be said. You should have that option in life."
Biden, whose son Beau Biden died in 2015, echoed Cuomo's sentiments, saying "How do you choose between leaving the bedside of your dying son or daughter who doesn't want you to leave, who just wants you to hang on?"
The United States currently offers 12 weeks of unpaid leave for family emergencies, but Biden said the rule doesn't leave people with many options when faced with the obligation to help loved ones in times of need.
"When we speak about the middle class, it's not a number in America; its a value system," he said. "It's not about a particular income, it's about being able to care for your family. It's about being able to look them in the eye and say 'honey, it's going to be ok."
About 40 percent of Americans are not eligible for the 12 weeks of unpaid leave to begin with, due to limitations within the 1993 Family Medical Leave Act. Many decide against the unpaid leave, even in times of emergency, because they wouldn't be able to support themselves and others in the meantime.