How do I remove back-ups from time machine?

Does that sound like a good solution?

No; that's the opposite of a good solution.

Can I use the AirPort utility to remove some backup that are over two years old?

No. As I wrote Time Machine does that automatically.

You have been given terrible advice that is 100% certain to leave you with no backups and a corrupted source volume. In other words, nothing. All your data are at risk.

The reason for deleting some back-ups comes from the storage on my MacBook is almost full.

Deleting backups will not change a thing about your MacBook. It will just leave you with a corrupted and useless backup.

Start from the beginning. If your source volume (the MacBook) is full or nearly so, then Time Machine may not be able to create the local snapshots it requires to operate. If you were to use CleanMyMac or any number of other bad ideas to "clean" those local snapshots, then Time Machine won't work. Stay away from those things. To reclaim sufficient local storage read the following Apple Support documents:

Optimize storage space on your Mac - Apple Support

Free up storage space - Apple Support

Unlike the alleged "genius" whose advice ought to be grounds for termination, those documents were written by Apple engineers who actually know what they're doing.

Next: If your Time Machine backup disk is too small to back up the contents of your MacBook then you need a bigger backup drive. The frequently parroted advice is to get one with two to three times the capacity of the source volume. Like most "rules of thumb" it lacks either mathematical or technical justification, but if you follow that advice it will certainly be sufficient. Apple says you should have a backup disk at least twice the size of the source volume: Back up your Mac with Time Machine - Apple Support. That should also be sufficient.

Next: You wrote that you are using an "old" AirPort Extreme. All Apple AirPort devices are "old" by now, but you should be advised that the only AirPort Extreme Base Station that is guaranteed to work with Time Machine is the latest model — the "tall" 802.11ac variety. If you are using one of its predecessor "flat" models your Time Machine backups are not likely to be reliable. Given Time Machine's mission profile you should only use a supported configuration. This site is littered with reports of people who used various unsupported Time Machine configurations only to find out their Time Machine backups didn't work in the dire circumstances in which they were required. Supported and reliable configurations include using a Time Capsule, the most recent 802.11ac AirPort Extreme, or a backup disk connected directly to your Mac.

Next: You wrote that you use Time Machine to create weekly backups. I strongly recommend that you let Time Machine run automatically, all the time. It will write those backups to the TM backup disk whenever it is available. That way your Mac will avoid having to create all those "local snapshots" that may or may not be contributing to the low storage space that you describe.

Finally: None of this has anything to do with creating space on the backup disk — your original question. Let Time Machine work the way it's designed to work. Provide a sufficiently large capacity backup disk (and preferably more than just one) and your problems are over.

If you use "CleanMyMac" your problems will have just begun...

<< At a Genius Bar appointment, the tech suggested that I delete some of the backups and then run CleanMyMac to see how much space could be freed up.>>

are you sure you were in an Apple store? if you were, that tech needs to be sent back for re-training.

Time machine keeps its own backups as neat as it wants them. if it needs more space, it consolidates older backups to make space. The only thing you can do by intervening is to corrupt the entire backup SET.

Do NOT use Finder to delete ANYTHING off ANY Time machine backup, ever.

You will also find that there is nothing to see in a backup set attached to an Airport extreme, because the backups are inside a sparse bundle disk image, and not visible in Finder unless you open the sparse bundle disk image, which you likely do not have permission to do.

--------

CleanMyMac is responsible for a LOT of badly running and corrupted Macs. It has been know to attempt to discard parts of MacOS as it is running. in addition, it does not correctly remove itself, which leads me to label it MALWARE.

Individual backups? You don't. Make no attempt to do that. Time Machine does it automatically.

Describe the reason you perceive a need to remove backups.

As John Galt stated you cannot remove old backups yourself as it would certainly corrupt Time Machine’s database and likely prevent you from ever restoring anything from those files. Time Machine automatically deletes older backups to make room on the Time Machine device when space is running low. If it’s not doing that then you have a corrupted Time Machine on your hands. Airport Utility is irrelevant.

Use the Disk Utility in the Utilities folder to reformat the dick. You can remove snapshots (called thinning) that TimeMachine has created over time. To read more about About Time Machine local snapshots - Apple Support

What TimeMachine does is creates backups until it runs out of room and then starts to delete the oldest backups and starts writing the latest backup.

ALL backups? usually, you just erase the disk.

If I recall correctly, you use Airport Utility for that.

If we are not talking about a Time Capsule, use of Airport Utility does not apply.