This is a problem of long standing.  Many years ago, Justice Thurgood Marshall said that the American people would not support the death penalty if they knew the truth about it.  The Marshall Hypothesis has not been borne out.  The other side has bombarded the public with their version of the "truth," and yet support remains very high, in the two-thirds region or so, depending on how you measure it.

The basic problem that the opponents have is that they cannot bring themselves to admit that supporters, which includes a large majority of the American people, really do have good reasons for their view.  They do not understand the viewpoints of people who disagree with them and make no real effort to do so.

This problem is not unique to the capital punishment debate.  For example, American liberals fail dismally in their understanding of conservatives, while conservatives have a far better understanding of liberals.  Jonathan Haidt demonstrated this empirically in his moral values research.  A portion of the sample was asked to complete the Moral Foundations Questionnaire to establish a baseline of how conservatives, moderates, and liberals answered the questions.  Then people from the various groups were asked to answer as they thought a typical person from a group would answer. 

The results were clear and consistent.  Moderates and conservatives were most accurate in their predictions, whether they were pretending to be liberals or conservatives.  Liberals were the least accurate, especially those who described themselves as "very liberal."

I understand why people who disagree with me on capital punishment hold the views they do.  I do not expect them to easily change their views.  I do not sneer at them with contempt and scorn for their views.  (I do have contempt for the anti-death-penalty activists when they lie, but that is a different aspect of the problem, and that group is a small subset of opponents generally.)  I see little or no effort at understanding from the other side.