Depending on how large the raised bed area is; if it is just a few raised beds in the area, putting cardboard down and covering with mulch would smother the weeds. If that is the case, I would recommend entirely covering all the way to the edges of the wood. Otherwise, the weeds will reappear at those edges. (Give 'em an inch…. ya know!)
If it is an extensive area with many raised beds, I would opt for a string trimmer to keep the walkways clear.
BTW I use a string trimmer to edge my flower beds and create a lengthy 'divot' along the area between the lawn and mulched beds.
Pretty dogs!
