Its not that we, today, do not believe in it. It was 1957 when the Territorial Government decided to not allow it. There was a history of potentially biased cases resulting in executions under "miners law" which resulted in natives and immigrants being executed after quick trials. Bills to establish capital punishment are introduced once in a while, but they never make it out of committee. The one issue that our society will have with capital punishment is the cost. Millions of dollars go to each case to house and defend the convict while waiting for the appeal process to finish up. Its cheaper to just keep them alive for decades. 

Copy and paste this link to learn some of the history.
http://justice.uaa.alaska.edu/death/ak_death_history.html