Thank you for your question and picture for Extension. Most literature says to wait until a frost, then cut your "fern" tops back to about 2 inches. It would also help to add 4-6" of mulch on top. Depending on where you live, you could do this anytime since we have been getting freezing temps in many areas of Minnesota. This article from Iowa State Extension says to leave the tops and keep the dead foliage to catch snow. It is contradictory to the cutting back method, which is so true of many gardening advice! Here is the article:

https://hortnews.extension.iastate.edu/faq/when-can-i-cut-back-my-asparagus-foliage

Here is a good Extension article about asparagus diseases and pests:

https://extension.umn.edu/growing-guides/asparagus-pest-management