I find the best, and least expensive, method of "weed" control in my Asparagus bed is mechanical removal and planting cover crops such as Field Rye and buckwheat. Since any "weeds" growing in what would be a new bed could add valuable nutrients to the soil I would simply turn them in as I was preparing that bed for planting.

There are several products which will help. Eptam is available for home use and will prevent many annual grass and broadleaf seeds from growing. If already growing when applied, it will have no effect on the growing weeds, though. It also needs to be worked into the top inch or two of the soil as soon as it is applied. A day or two of direct sunlight will break it down to nothing. Any weeds growing prior to the emergence of the first spears in the spring can be sprayed with glyphosate. Products with 2,4-D will control broadleaves once they are growing if applied right after picking when there are no asparagus spears showing. 2,4-D will cause the asparagus to curl if you do get it on any growing spears, and while it doesn't kill it, it certainly holds it back for a while. There are also several herbicides available only for agricultural use that will provide good control of weeds in asparagus.

If you get at it while they are small and the ground is moist, a light hoeing in the spaces between the crowns will also do an excellent job of weed control. I don't find hand weeding among the plants to be that bad when the ground has good moisture so the weeds will pull easily. Later season weeds don't usually get too far if you have several rows of plants as the ferns create a dense shade on the ground and choke the weeds out as long as they were controlled early and not allowed to establish until after the ferns have created dense shade.

I generally prefer to make a herbicide application before the first spears emerge, then do some hand weeding while picking to remove dandelions and other perennials.

How about salt? For healthy asparagus, add sodium chloride rock salt (NaCI) to beds more than one year old. Apply 2.5 lbs. per 100' row before spears appear. This treatment prevents crown and root rot diseases caused by fusarium fungus and also improves overall growth. Don't use iodized table salt or rock salt made of calcium chloride (CaCI). Pickling salt is fine. (re: Rodale's All-New Encyclopedia of Organic Gardening).
A lot of competitive weeds can't handle salt. HOWEVER, this is a product that will leach into surrounding soil. Good stuff may be affected if you use it year after year...