How about a button that says something like "Set order", and then displays the message "Click actions in the desired order of execution (you can always change that)". The user clicks through the list, with an indication of the number of the current step ("Set step #7"). The order is reflected by numbers on the actions. When they click "done", actions are rearranged in the right order. From there, up/down arrows and drag & drop should work well for adjustments.
Letting them edit the numbers manually can turn into a major headache for everyone. Let's say I want to switch between steps 5 and 9. How do I go about it? If I type "9" in the box of the current 5, then for a second I have two steps number 9 and no step number 5. If I delete both values first, then for a second I have no steps 5 and 9. Of course it can all be solved if you make them enter a special "edit" mode with validation at the end, but it just makes everything much more cumbersome, and the complexity of your UI just went up a couple of notches.