Karen;

You Could redo the bed. By taking out the "new" rosettes of
the strawberry plants, than store them over winter in a cold frame till spring- with the rosettes removed you could lay black plastic over the existing bed thus killing the grass until spring planting time for your berries.

I agree with previous poster. I prepared a new bed, then moved the 'rosettes' over this past spring. The old bed got 'lasagned' about 18 inches deep, topped with bags of topsoil, and planted with veggies. Very little grass has come back through, and what did was easy to pull. I mulched the new strawberry bed with oat straw.

Get on top of it by hand weeding and keep it up. I keep two 50' rows weed free with an hour or so a month.

If that isn't possible use a chemical called Grass B Gone.