Fabric to wood adhesion is commonly used in furniture making. It can be stapled or glued to the wood, depending on the piece of furniture. Typically, the best looking fabric to wood method is aerosol trim cement glue. Aerosol glue is extremely sticky and won't saturate the fabric or the wood. It stays sticky forever and can be surgically applied to seal down large panels of fabric, or glue down loose corners.
Spray a light coat of adhesive over the wood. Don't let it pool up, just spray an even coat on the wood. Let it cure for three minutes.
Hold the end of the fabric up, so that the end of the fabric is square with one edge of the wood. Let the fabric overlap the edge by one inch. Begin laying the fabric down on the wood. The edge of the fabric will instantly bond with the glue.
Hold the fabric up high with one hand, as you press and smooth the fabric into the glue with the other hand. Lower the fabric; feed the other hand loose fabric to smooth and flatten, as you move across the wood.
Smooth and lower carefully. If you start to get a wrinkle, pull up on the fabric gently, and then smooth it out with the other hand.
Smooth and lower across the wood, until the fabric is completely bonded to the wood. Trim the edges of the fabric off with scissors.
Cut your fabric at least two inches bigger than the wood. It should overlap the wood on all sides.
If you get adhesive on something other than the wood, you can remove it with acetone. Always check the nozzle of the glue to make sure it's pointing in the right direction before spraying. Wear breathing protection when using aerosol glue.