Certified GKH Gardening Expert
Some people do use plastic, but that is more to keep deep rooted weeds from growing up through the bed. If you decide to do this, be aware that the plastic can restrict the roots of your plants. It is probably not needed unless you had a serious weed problem where the beds are now.
Chicken wire buried around the beds is an excellent way to keep digging critters out of the bed. It will not harm your plants. You will want to bury the chicken wire 1-2 feet to be effective against digging animals.