My guess is that you have a deer problem, may be some other animal. If deer, you can attach 4 foot high 2"x4"s to your existing base boards. Then attach welded wire to the outside of the 2x4s, welded wire is the 1/8" wire strung with 2"x4" spaces-much stronger that chicken wire. String some of the welded wire between 2 of the diagonal corners to support growing cucumbers. Make sure you have a method to access the cucumbers.
Deer do not jump fences that have some thing that will result in their injury. Learned this by stringing 1/4" twine to the top of a 4' high wire fence of a veggie garden. Deer did come in, but only when I left a 3' or more spaces between the twine which allowed them to land without injury.
