There are a some herbicides that can be applied before the asparagus emerges.
In the past I have used Preen.
To catch up with a really weedy bed, I started in very early spring with an application of Preen.
After a few days, I raked it clean, and then I went over it with the rototiller.
I knew my plant roots were 12 inches down, so I felt safe tilling to a depth of 4-6 inches.
Waited a few days, did some more Preen on new weed growth, waited a couple of days and went again with the tiller.
After that season, each spring, I always went over the area with the tiller, but very shallow---and then if needed, I used an application of herbicide----at least a couple of weeks before the asparagus was due to emerge.
By fall, my bed would be weedy again, but , come spring, I'd do the preliminary prep as described, and I always had a good healthy harvest.
I had started that bed from seed, and it was still a good healthy stand of asparagus when we moved 25 years larer.
Now, I'm just getting started with a new bed at our new home.