Fish that are to be returned should be handled with wet hands for as little time as possible and returned to the water for recovery. With larger fish one should avoid lifting the fish as this can damage the internal organs. If you are not intending to return the fish then it doesn't matter how you handle it as you are going to kill the thing so it can fill up that space at the back of the freezer for the next three years. If you do kill fish, do it so as not to upset or incense non fishing onlookers. We are all ambassadors for the sport.

Fish breath via gills using the dissolved oxygen in the water to exchange the CO₂ that builds up in their blood. During the fight, the fish produces more carbon dioxide, as it's normal level of exercise has increased. Once the fish has landed the oxygen content in the air is greatly increased in comparison to the water that the fish usually breathes in. This initially gives the fish an extra rich source of oxygen that can help in reducing the dissolved carbon dioxide in it's blood. However, this enriched oxygen environment very soon becomes poisonous for the fish just as humans would die if they were to breath air with a high oxygen content, which is opposite to the usual theory that fish 'drown' (lack of oxygen) if they are kept out of water. They in fact suffer from oxygen poisoning due to the over rich oxygen environment that air provides. Once the gills dry out, the oxygen transfer becomes very inefficient and the fish then drowns (if it is not already dead).
Fish should be handled as little as possible out of water and placed back in their watery environment as soon as is possible to prevent this poisoning. Larger fish can suffer internal injuries if lifted from below due to their large size being supported in the two places where the hands are. They usually have the pressure of the water pressing on their flanks keeping the fish in correct proportion. Once this pressure is removed large fish are already struggling due to their increased weight (they only weigh 1/6 of their true weight under water) and if this weight is concentrated in only two small places where your hands are then you are in effect increasing the weight in that area even further.
Wet hands are used for handling the fish as fish have a protective slime on their skin that protects against parasites, disease, abrasions, etc.
Dry hands will strip this slime very easily and make the fish more susceptible to disease. Fish can quite often be seen swimming in 'Put and Take' lakes with hand shaped fungus growths on their sides from poor handling in the past.
For this reason it is also important to use wet, small mesh, knotless nets when landing fish that are intended for return.