My iMac no longer recognizes my backup drive

I'd certainly try a new cable first. Drives do fail, though. Are you looking for an HDD or an SSD? The latter are much faster.

Is there a reason you need the drive itself to offer encryption? It's quite easy to format a new drive with Disk Utility and apply strong encryption:

Encrypt and protect a storage device with a password in Disk Utility on Mac - Apple Support

If you'll be using Time Machine to back up your Mac, don't encrypt the drive itself but rather select the option to encrypt the TM backup when you set it up.

So when you say “my backup drive” do you mean Time Machine or third party backup software?

And yes, you should try a different cable before deciding to buy a new drive. You might also try exercising the existing cable by unplugging and plugging at both ends a couple of times. Corrosion can and does muck things up at times.

You might also try disconnecting your backup drive, restarting your iMac, and then plug the drive back in to see if it shows up.

As for a recommendation all of my external drives were purchased from OWC/MacSales. You do NOT want to use any software that comes with a drive, especially password protection. You are begging for trouble. macOS 15.6 Sequoia Time Machine will reformat any drive you set as the Time Machine drive to a set format that is read only to the user.

Does the drive have its own power supply or do you just connect it to the Mac?

If it doesn’t have a power supply, it may be drawing too much current on the USB and shutting down the port.