Router console ports are meant to allow root access to the router via a dumb terminal interface, regardless of the state of the router (unless it is completely dead). By connecting to the console port you can get remote access to the root level of a router without using the network that the router is connected to. This creates a secondary path to the router outside the bandwidth of the network which needs to be secured without relying on the primary network.
This allows a technician sitting in a Network Operations Center thousands of miles away the ability to restore a router or perform an initialization configuration securely over a standard telephone line even if the primary network is in failure. Without a connection to the console port, a technician would have to visit the site to perform repairs or initialization.
In short, it's nothing you need to be worried about.
Feedback received on this FAQ entry: