The perspective of the question seems inverted. We don't have borders per se, because we like to cut through things. We have Countries that exercises their authority on an specific area. The perimeter of that area is called a border.
The area that each Country control is limited fundamentally because there are disagreements on how to best run the thing or who should run the thing. If ayouyou allow everybody to enter your Country you lose control of it and bad things may happen: economic troubles, political crises, loss of political authority, violence, etc.
Just like many people like to have their own houses, many people like to have their own Country.