Admittedly, I oppose the death penalty in the vast majority of scenarios we're likely to encounter. I'd support it for the Joker -- dude breaks out of Arkham every other week and murders at least thirty people every time he hits the streets, so killing him would be in the interest of public safety. But insofar as our criminals are contained and aren't continuing to menace the public at large (by, say, running a criminal empire and ordering hits and the like), I don't support killing them. I maintain that killing people who are no longer a threat diminishes us. (And, of course, there's the possibility of condemning innocent people and the reality of disproportionate application of the death penalty along racial lines and all of that jazz.)
In any case, whether one supports the death penalty or not, I hope that most of us could agree that there should be some discussion whenever we seek to apply it -- that we should consider what we hope to achieve by it and whether it upholds or runs counter to our values and aims in particular instances. Might it actually be a deterrent in this case? Will it instead make a martyr of a perpetrator and inspire others of his ilk? Could it deprive us of the opportunity to learn more about the perpetrator? Insofar as we could convince him of the error of his ways, could it deprive us of a potential ally? And so on. In short, whatever one's feelings about the death penalty, it should not be taken lightly.
It should definitely not be the kind of thing a US president demands, in all caps, without qualification or apparent thought.