Roundup will kill your asparagus.

Crabgrass is an annual. It will die off with the frost in the fall. Next spring, you can apply a pre-emergent to keep seeds from germinating in your beds. There is a version of Preen for vegetable gardens.

How mature is the asparagus? If its more than a year or two old, you should be able to mulch relatively thickly around/between/under it and drown out the crabgrass without harming the asparagus. I'd suggest a roughly 4 inch deep layer of either clean straw. This is what I do with mine, and it seems to really help the asparagus grow as well. As it rots, it enriches the soil.

An alternative would be the selective herbicide "Ortho Grass-B-Gon Garden Grass Killer" -- I have found this very effective in killing weedy grasses, and is safe to overspray on most plants. However, just to let you know, I have NOT tried this on asparagus. It doesn't harm other members of the lily/asparagus family (families? -- the botanists are ALWAYS changing things around, just to confuse us I think!) that I have used it on/around, including Asiatic lilies, hostas, solomon's seal, and many spring bulbs like fritilarias, hyacinths, etc. Technically not labeled for use on vegetables, but ... you can make up your own mind about that issue.

Preen would be a good idea next year.

Here is a link that might be useful: Ortho Grass-B-Gone Garden Grass Killer

"Or I could just let it grow and worry about it next year."

I can not recommend that procedure. I tried that last year!

I had such a problem getting everything planted (two feet of rain between thaw and normal planting time), that, by the time I got around to dealing with weeds, it was too hot for me to expend the energy to clean everything up. (I have a heart condition, pacemaker, and can not do much "heavy lifting", or even steady work without very frequent rests, and working very hard in 100F temperatures is not possible.)

It was just so hopeless out there with crabgrass in my asparagus (the neighbor on that side of my garden uses crabgrass as the variety of choice for his lawn), I let it go.

I have just been greeted with the results of that: crabgrass sprouting everywhere! I had a drier spring this year, and had almost everything planted on time, and have had cool enough weather that I have the crabgrass and other weeds under control. I used the hoeing and pulling method, but I will use some pre-emergent in the future.

If it gets too hot and I keep having more crabgrass sprout before I find the pre-emergent, I will use "roundup". I will use a cheap paintbrush and paint the seedlings with it.

I am not afraid of roundup. I was able to defeat it in my bluegrass lawn by painting some encroaching sprigs of it with a small artist brush using a "roundup" substitute, and the space I garden in now, was the backyard of a neighboring house that had a Bermuda grass lawn that I sprayed several times over about a two month period the year I bought it (and a few sprigs sometimes for the next several years.

I have not had any appear for quite a while now, and never killed any desirable plants or lawn grass with it.

I don't know that Grass-B-Gone will work on the crabgrass. It's pretty specific.

I've been beating back crabgrass in some areas. Here's what I've done. I've put down as much cardboard as I can get my hands on, and then 4" of mulch. It's year two on this one patch, and so far the nasty buggers haven't come back. Of course, it looked like that last year at this time, too, so maybe I'll get some next spring again. So long as it's the right direction (less crab grass every year), it's OK if this takes a while.

I get my cardboard at a local appliance store. This year, the nasty crabgrass is topped by a triple layer of washer/dryer boxes and then mulch. Beat that, suckers!

It's pretty fast to lay down, and I think it works better than landscape fabric. Crabgrass jus laughs at landscape fabric in my yard. Good luck!

DO NOT let it go to seed.
Mulch the bed with leaves or straw, old bed sheets, anything that will make the grass rot.
Here is a link on how to care for & extend your harvest of
Asparagus.

Here is a link that might be useful: Clemson