This bugged me for different reasons. The first police officer who saw her was calling an ambulance. She refused. Gave him a note to deliver. He delivered it to the Police Commissioner, who stopped what he was doing and went down to the docks to give it to Batman.
And yet this whole thing seems to be making the point that Batman saved her and everyone's against them, because they are outsiders who live by their own rules and have to stick together against the cruel unfeeling world.
So they get to treat public officials as their servants and still revile them for not being helpful or effective. I guess this is the film noir version of having your cake and eating it, too.