I just leave mine. They'll die down themselves, and you can take the dead leaves off when they're dead.
I move the pots to a bit of shelter for the winter, and tidy them up in spring ... I give them some fresh compost then aswell.

:o)

Maybe put a label in the pot so you remember what's in there. Also add just a few slug pellets in Spring to kill any hungry slugs and snails.

I have hostas in pots to but 2 of mine are do not have drainage holes, they are snail shell shaped. These get tipped on their lip and brought in from the danger of freezing.
so if the pots are expensive or special then that needs taking into consideration.
when the plants die down I remove the dead foliage, just like Hywel says.

In the spring they get fresh compost and slow release fertilizer.

I leave the leaves on any plants while they are at all green to strengthen as I think they put strength back.

In the case of hostas I clear away dead foliage at the end of the season, and divide them in spring if they start to get too big for the container.

I find them fully winter hardy (in Devon anyway). I had a huge slug/snail problem so I grow them in hanging baskets, which works very well as long as no part of the plant gets near the wall to provide a bridge.