Killing criminals does not stop crime, either as a deterrent or by removing criminals from society so they can't reoffend.
Execution doesn't work as a deterrent because anyone who commits a crime does it thinking they wont be caught, or (well known phenomenon) doesn't think any further than carrying out the crime itself anyway. Some don't even care if they get caught. This one is a no-brainer, and statistics support it. The only way to make deterrence work would be to make policing 100% efficient (which would require, say, two to three policemen per citizen), which is impractical, and also to make sure nobody is dumb enough to think they wont get caught even if there is 100% effectiveness, which is bloody impossible given the general stupidity of a colossal proportion of society.
Execution doesn't work by removing criminals from society because all criminals were previously perfectly law abiding for some finite period after their birth. In order for execution of criminals to halt crime, you would also have to prevent any further births (impossible) and maintain total social stasis (also impossible) so that everyone who currently isn't a criminal doesn't encounter some new incentive to become one.
Since neither deterrence nor prevention of reoffence by removal are valid justifications for execution, what do we have left but the concept of justice, which is basically a more acceptable name for revenge (for this reason, I always feel uneasy when I hear people calling for "justice"). Is satisfaction of vengeful citizen's bloodlust and hatred, without actually improving society in any way with regard to reducing crime and maintaining fairness and order, sufficient justification for the state to murder people? And let's be honest here, how could killing only in the name of vengeance be called anything but murder?
The law should be a matter of enforcing fairness and protecting rights. The law should not be a matter of revenge. If a right has been lost and can be restored, go for it in the name of fairness. If a right is irretrievably lost, taking the same right from others just makes the net situation worse.
