If any archers did stick their arrows in the ground in front of them (a common sight in fiction, but less evident in actual medieval depictions of archers), it would definitely be for purposes of convenience rather than any notion of infection. For one, medieval understandings of medicine and disease were not the same as ours today, so dirtying up an arrow to try to give someone tetanus would probably not have been within the understanding of Ye Olde Footsoldier. Even if they did have a modern understanding of infection and disease, giving the enemy infected wounds wouldn't have been a priority for them. The purpose of archery on the battlefield was to make the enemy's life more difficult, by killing/wounding/harassing them and making them less able to carry out their plans. If an archer nicked someone in the left elbow with a dirty arrow, thus causing an infection that kills him three weeks later, that wouldn't really be of great benefit to him on the immediate day of the battle, because it doesn't do all that much to prevent the enemy from running said archer through with a sword. The archer would primarily be concerned about his ability to access his arrows quickly and easily rather than what may or may not happen to the enemy days or weeks after a fight.

In general, medieval archers both in and out of combat are depicted carrying arrows from their belts somehow. Sometimes this arrangement is as crude as shoving a handful of arrows directly through the belt, but more sophisticated strap systems or knots that hold larger numbers of arrows stably are also quite prevalent in artwork. Arrow bags/sleeves are also very common, but generally hang from the belt rather than over the back like in Hollywood.