Everyone needs a passport for Alaska, including Canadian and US citizens.
With a criminal record, it would be the same conditions as crossing the US border by car. If you have issues there, there may be issues with boarding the ship as well. Therefore, we cannot guarantee anything and neither could the cruiseline because it's a customs/government matter. So it's best that you visit a US consulate to ask these questions about border security and immigration. This way, you can be worry free during your trip.