I’ve already written at length about the longstanding Catholic teaching about the death penalty and why Pope Francis is wrong in his attempts to change it. I won’t, therefore, rehash those arguments here.
For the purposes of this particular debate, what matters most is for Catholics to understand that the authority for civil powers to use the death penalty exists, that its proper use is not intrinsically evil, that this truth has been divinely revealed, and that it has also been magisterially affirmed. This is, therefore, an infallible, irreformable teaching of the Church.
For those who would like to see more theological substantiation, Dr. Edward Feser, who co-authored a book on this topic with Dr. Joseph Bessette, argued persuasively in an article last year at Catholic World Report with specific reference to statements made by Pope Francis. Dr. John Joy, who wrote the best explanation I’ve ever read on the proper understanding of the modes and exercises of the Magisterium, also wrote in support of Feser’s argument.