Is there a device that has multiple serial ports that I can connect to my computer so that I can connect other networking devices that can be configured from the serial port to this device and then configure these devices from my PC?
Here is a diagram of what I am trying to accomplish. I am looking for a device that I can use in the red position in the diagram.
You are looking for a serial terminal server or console server. Many manufacturers make them such as Raritan, Lantronix, and Black box.
As @Bert said, one way to do this is with a Serial Terminal Server or a Console Server.
However, if you will only be using one of these devices at a time and you will be in the same physical location, you could use a much simpler physical switch box, like this one:
These devices usually have a bunch of buttons or a dial on the front and they actually physically switch between the different connections and your computer.
It's much more old-school, but much easier and cheaper if that's all you need.