Do I need to worry about rain water getting into the open ground socket?
You would do well to pull some slack to that extension cord and arrange the plugs so that they at least face down. It is not just the GND terminal of the extension cord that is of concern. You also have the open prong holes on the stackable Christmas tree light string that is facing up. 

Water that gets into the electrical plug contacts raises the most concern that it can and will lead to corrosion on the metal contacts.

I have found that a better overall solution is to arrange the Christmas light installation so that plugs and extension cords are terminated up under the roof overhang.