When Dodgeson is getting out of the cab, the driver gets out at the same time to shut the door. If Dodgeson really left it open the cabbie should have realized this after he walked away and not as he was getting out.
I read this comment while this part came up on the movie and the cab driver got out when Dogson stood up to get a tip. That is why when he walks away the cab the driver sticks his hand out.
We cant even really see the cab driver til he is on the Dodson's side of the car. He comes around as Dodson is walking away to shut the door and gives an annoyed gesture behing Dodson's back.