What you are proposing will do nothing to increase safety. The safest place for a weapon is in its holster, and constantly having to remove it to enter a "Gun Free" zone will only decrease safety. Futhermore, creating permanent gun-free zones on state property will endanger the lives of everybody in those zones. Keep in mind, the San Bernardino shooters sought out government facilities to do their killing, knowing that those facilities are gun-free. Why in the world would you want to place a Virginia citizen, and state employees, in the same defenseless position?
This rule is not a reaction to any need. It is simply the Governor's attempt to further his own anti-gun, anti-freedom shibolleth.