I take a small spray bottle with a direct jet, not a fan spray, and douse the bermuda clumps (not runners, but where they are rooted in) with Round-Up, being very careful not to get any on the asparagus. I have to do this three or four times a year, and pull the runners out by hand. Bermuda is our bane, but the asparagus sure is good!
You are so lucky to have "inherited" an established asparagus bed. It will be worth the effort you put into it!

You can also apply it using a cloth or sponge-like material with the RO soaked into it by applying it to the leaves.

Wear gloves or put it on some kind of stick.

or paint it on with a small childs paintbrush, tedious yes, but you'll only get round up on the grass.

Wait until the ferns die back brown. Cut them to the ground. When you get the first nice weather in February or so then spray the whole bed with round-up. It Will Not hurt the crowns. I've been doing this on my 1000 crown bed since '99. Tom

It took two or three years to get it all but I eliminated Burmuda grass from my asparagus beds by heavy mulching. After the gus died, I cut stalks and removed them, then covered entire area with 4 to 6 inches of shredded leaves.
The asparagus stalks pushed through much better than the grass and I pulled sprigs of grass that did peek through. In addition the rotted leaves provided fertilizer the following year. You'll need to add lime ever year or so though since composted leaves are highly ascidic.
I have read, but not tried, that sprinkling salt (plain sodium chloride) will not harm asparagus but will kill most other plants.

Asparagus does do well salted. However, if you have deer, they will excavate the whole bed, turning it into one giant salt lick.

Brook