I successfully propagated a couple of papaya trees with cuttings, but don't think it's worth the effort. I cut off side branches on old papaya, including the swollen knob at the base of the branch. I removed all the leaves except the growing tip, planted in a sandy soil with compost added so it retained moisture but had good drainage. They were kept in mostly shade for a month or more.
Papaya fruit so fast, and seedlings are relatively true to the mother fruit, so there's little to gain from using cuttings.