My significant other attached clothespins to the picnic table by first taking the clothespins apart then drilling a hole through the clothespin. Then he put a screw through the drilled hole and screwed the clothespin to the underside of the table. Then he reassembled the clothespin. Below is a pic before he reassembled the clothespin.
I bought two tablecloths from Walmart and cut one to fit the top of the picnic table and the other one to fit the seats.
I attached the tablecloths to the picnic table with the clothespins. You can use a lighter to burn off the frayed edges after cutting to that the cut parts of the tablecloth don't come apart.