The affect of weeds on asparagus and other plants depends on the weed.
God forbid something like field bindweed get a foothold and take over...it's an evil plant.
Where I'm at, Ground Ivy AKA Creeping Charley is a very serious problem.
It can smother asparagus.
ETA: Lambsquarter and patience dock are edible weeds, but I wouldn't want them in my asparagus beds...they can also take-ever.