We spent a week there in January last year and saw no one crossing the river in the park. Much of the river adjacent to the park is in a canyon, so would be very difficult to climb down the cliffs to cross.
We were there in the cool season (there is no "cold season"), so had 60 degree days. But Presidio is located just a few miles upstream from the park and often has the highest summer daytime temperature in the nation. So, you would likely have more problems with heat stroke than crime if you are going to be in the park at that time of year!