I had rooted some mint cuttings in water but the upper ends of these had already wilted a bit. They have rooted and I have placed them in a 10 inch pot with potting soil. Yet due to wilting the tips are touching the surface of the soil. Should I support it with some bamboo sticks? Or do something else?
The real cause of the problem was probably that your mint was already wilting and dying before you put it in water.
If the end of the cutting was wilted when it was still in the water, either it wasn't taking up enough water to prevent wilting (would would probably be fatal in the longer term) or else the end of the cutting was already dead.
I would have left the cuttings in water until there were new shoots growing from buds at the leaf joints, without wilted ends. Then you could prune off the wilted part before potting the cuttings, and avoid the chance of disease entering the wilted part from the compost.
If a cutting in water is not taking up enough water to stop it wilting, putting it in compost is not going to make it take up water faster and fix the problem.