The ability to alter the values has been removed. Adjusting the values was removed in 2.69. Up until 2.71 you could adjust the camera only option.
While you may not be able to adjust the values from the operator you can access and adjust the values used for the border. For the final render they can be found at
scene.render.border_max_x... Note that min to max values are 0.0 to 1.0 so the clipping is a percentage of the render size. You should be able to alter these values in a frame_change_pre handler.
I think cropping the image in the compositor would be a better approach. You can keyframe the settings of a crop node to move it around.
Some other options are to animate a mask in the movie clip editor (part of the camera tracking) or use object pass index and animate an object to create the mask.