How can I keep copper door handles clean?

You can spray with spar varnish one cleaned to your satisfaction. Best to remove from door.

Brite Boy works good on copper.

Problem is most of the time entry locksets are coated with a clear sealer at the factory. Bar Keepers Friend probably rubbed the some of the finish off. That's why it's blotchy. Try using ketchup to clean the copper. Slather it on and let it sit for a while then wipe off. Seal with Rustoleum 2X spray paint. Tape around the faceplate so not to get any on the door.

Your door hardware is not solid copper, but only copper-plated.

From the picture you posted, it appears that that coating has worn off.

I would not think that it was the cleaner (was it BarKeeper's Friend ?) that caused the coating to come off, but more so simply wear and tear over time.

It looks like whoever lived there before had a big bunch of keys and scratched the finish off every time they unlocked the door. You may need to removed the latch and lock and spray paint the whole thing, or just replace it.

I agree with all above. It has a copper finish that's simply worn off. Door handles and knobs get a lot of exposure and take a lot of wear and tear. If you can't afford to replace it, the best thing would be to remove all the hardware and repaint it with some Rustoleum Copper 2X. The can should say it has primer in it, too. I love this stuff. Finish with your BarKeeper's Friend to prep the surface. Follow the directions and make sure you don't clog the keyhole as you spray. Three light coats should do it. Follow that with with an exterior clearcoat spray - whatever finish you like: satin, gloss, etc. Replace the lockset and remember what you did. You'll be doing it again in a few years' time.

Here's a tutorial that might help - https://chascrazycreations.com/copper-cleaning-hacks/