Please refer to the Midwest Vegetable Production Guide for Commercial Growers - 2019
Here is the asparagus chapter:
The weed control section starts on page 89 and goes through page 92. The table on page 92 is particularly useful.
If you know which weeds you have, then this table on page 68 may also help guide your decision: https://ag.purdue.edu/btny/midwest-vegetable-guide/Documents/2019/ID-56_2019.pdf#page=70
If you're not a commercial grower, and this is just a garden patch, then the most cost-effective thing you can do is pick the patch clean after harvesting what you want this spring. Leave all the weeds. Then, spray the patch with glyphosate (Roundup) before the spears re-emerge and fern out for the rest of the season. Later, sometime in August or September, mulch the bases of the plants heavily to prevent winter annual weeds from germinating. Some perennial weeds will still get through next spring, and you can use the same clean pick and glyphosate method. Continue as needed.