Next year's asparagus depends on the healthy growth of this year's ferns, which should grow into very tall plants and last until the first frosts. The time to cut them down is at the very end of the season, or even early winter.
You may notice that the leaves are affected by small red beetles, the asparagus beetle, which lay eggs which hatch into disgusting grubs which look like droppings, which then munch through the ferns.
If you see the beetles, catch and crush them (a cupped hand BELOW the pest allows you to catch them as they jump off the plant when approached.) Just squash them between finger and thumb. The grubs can be cleaned off with your fingers too if you are not squeamish, or you can spray with an insecticide.
If you don't remove asparagus beetle the plants can be seriously damaged and the stems can rot off at soil level.
Before winter, or in early spring, cover your asparagus bed with a deep mulch of well rotted compost. This will both feed the bed for the following year, and also makes it easier to cut the new spears as they push through the mulch.

Oh thankyou Bertiefox for explaining all of that to a guest ! Just what I needed to know.