Parking on the river front is $8.00 for all day for the Gateway Arch Park grounds and for the other riverfront attractions like the Mississippi boat cruises and the helicopter tours. To get to the parking lot you need to take Chouteau Ave. to Leonor K Sullivan Blvd. and turn left. This will lead you to the parking kiosk.
Here is the Google Map for parking. This parking lot is right on the riverfront so there isn’t a lot of walking involved. I do believe this is the closest parking option available. Please remember don’t go anywhere downtown alone so bring someone with you if you decide to go. Stay safe!