According to Greek historian Herodotus, Persian Emperor Cambyses II ordered Sisamnes skinned alive for delivering an unjust verdict after accepting a bribe. He further ordered that the unfortunate judge's skin be used to upholster the chair from which his successor -- Sisamnes's son -- would hear future cases. A small vignette in the upper right-hand corner of David's painting shows the son dispensing justice in the macabre chair.
Executions just create more violence. Someone is even getting paid to kill. Someone else is really angry that this person was executed and might seek revenge, thus creating a cycle of violence.
Counterpoint:
For like 1000 years (Dark and Middle Ages) Europe executed like 1% of it's population every generation for crime. Then it conquered the world, and we have far less violence than we used to (Steven Pinker writes about this).