Landscaping fabric, the toughest you can buy . I try to produce ridges in the prepared soil, apply the fabric, pierce with a stanley knife, as if a hot cross bun , and carefully plant along the ridges. Try not to get any loose soil on top, bury the edges and overlap if joints are necessary, making the joint on a ridge.
These fabrics prevent most weeds from growing up from beneath but allow weeds to become rooted and established in soil left on top. Roots will penetrate down, causing difficulty in removal. Strawberry runners ,if left alone, will also get through, and create a jungle!
Strawberry beds have a realistic life of three fruiting seasons, so its not a bad idea to take runners each year, from the healthiest plants, and extend the bed by 1/3 each year, moving across the ground, like a continuous migration/rotation.
The oldest bed , having had lots of manure at the start, should be incorporated back into the general rotation and that means potatoes for me.