Here was my plan. I was going to put soundproofing foam between the rivets in the ceiling and line the 1'x6" rivet too. I was also going to line the walls of the staircase leading up to the door out. I was going to line both sides of the gate, the walls and ceiling of the 3x3' space between the gate and the door and the back of the front door with foam as well as ceil the door by lining it with acoustic seal and putting a door sweep on it.
Foam can not sound proof in any way. If anything it will absorb upper frequencies inside of the room but that is not going to stop sound from getting out of the basement. Soundproofing takes mass and can be very costly.