A witness said he saw someone throw something into the bus
The blast happened as passengers passed through a security checkpoint before boarding a coach, which was bound for Kampala in neighbouring Uganda.
The person who died was carrying a bag which contained the bomb, Police Commissioner Matthew Iteere said.
Earlier, Uganda’s police chief had warned of attacks by Somali militants.
Most of the wounded were Ugandans travelling home for Christmas, Red Cross official Nelly Muluka told the Associated Press.
I felt like I was hit by an electric shock” End Quote Jack Wachira
The blast smashed the windows of the Kampala Coach vehicle, and left blood stains and scattered bags on the ground nearby.
“I felt like I was hit by an electric shock,” Jack Wachira, who had escorted a friend to the bus station, told the Associated Press.
“When the particles hit my leg, I thought I stepped on electrical wires. Then I heard the explosion.”
Mr Iteere said it was not yet clear if it was a suicide attack.
The BBC’s Will Ross in Nairobi says the fact that the bus was headed for Uganda has certainly increased suspicion and the region has good reason to be nervous.
Last July, the Somali Islamist militant group, al-Shabab, said it was behind the twin bomb attacks in Kampala which killed more than 70 people.
The group also warned that there would be more attacks so long as foreign peacekeepers remained in Somalia.