For shower curtain, to avoid damaging the screen I would use some magnets. They won't stick to the screen so you would need some that you can stick both inside and outside to hold each other up. You should be able to find magnets at home improvement stores, on Amazon, etc.
You could also wrap the curtain around 3 sides, tuck the bottom under the cage, and weigh it down on top.
There are other methods but personally those are my favorites.