Issue with connecting external hard drive to AirPort extreme

Although the Airport Extreme has a USB port that is only for printing.

This is not correct.

Reference this Apple Support document:

Share a USB hard disk on your network in AirPort Utility on Mac - Apple Support

If you’re using an AirPort Extreme: You can connect an additional USB hard disk to the USB port on your base station.

Assuming you are talking about an external hard drive, not the hard drive on your Mac or the hard drive on your Airport extreme..

Although the Airport Extreme has a USB port that is only for printing. It won't work for anything else. So if your hard drive has an ethernet port then just connect it to the Airport and it will appear on the network. If it's got a USB port only then you'll probably have to buy a USB to ethernet adapter, but I've never used one so you'll need to do some research before buying to make sure it will work on your network and you understand how to configure it. Don't just buy the cheapest one from Amazon - make sure it's designed for adding storage.

How is your hard drive formatted?

It will only work if you have formatted the drive in Mac OS Extended (Journaled), because that is the only Mac format the USB port on the AirPort Extreme will support.

Keep in mind that the USB port on the AirPort Extreme is underpowered, so you may need to use a hard drive that has its own power supply.

Did you configure the Extreme to allow disc sharing?

Share a USB hard disk on your network in AirPort Utility on Mac - Apple Support