Similar question discussed just a bit further down the page but no, you do not cut them back until they brown and die this Fall. Otherwise your crowns die. Your plants are doing exactly what they should. Keep them well watered throughout the summer and feed the bed well in the Fall and again next spring.

You can frame the bed in with posts and string to help them stay upright if you wish or if the weight of the plant appears to be pulling the crown up out of the soil. But it isn't required.

Dave

https://www.gardenweb.com/discussions/5326323/asparagus?n=6

Just wrap some string around them and cut them in the fall when they turn brown. Nancy