Thanks for writing. Your best bet would be to cut the grass as low as you can, then cover it with several sheets of newspaper and then cover that with a couple of inches of mulch or compost. Then dig a perimeter around your asparagus bed and edge that so you don't have grass creeping back in.
Some grass may start to grow through the mulch --- if that happens, pull it as soon as you notice it.
---Justin, Senior Garden Editor, BHG.com"