"We must complete this mission and we will," he said. He also vowed to hunt down the perpetrators.
More than 100,000 people have been evacuated from Kabul, which fell to the Taliban on 15 August.
But many more Afghans have been rushing to the airport ahead of the 31 August deadline for US forces to leave.
Mr Biden vowed to complete the mission, adding "we will not be deterred by terrorists".
Thursday's attacks happened at about 18:00 local time (13:30 GMT).