My plants are growing all over and now into the driveway. If I cut the lead edge will that harm the plant or just stop the growth?
You can locate the main stem of the plant and remove any side shoots that are growing from this main vine.
Remove the first 4 to 6 shoots growing closest to the base of the plant and leave all the rest.
Plants grow quickly, so you may have to do this more then once.
I also may suggest putting up a temporary fence or trellis to gently place the cucumber vines on.
This article will help you.
For future planting, I would suggest growing a 'Bush Cucumber'. I have raised beds and this type of plant stays much more contained and still produces lots of cucumbers.