Competence Breeds Trust
If you’ve run audio for very long, you’ve no doubt encountered timid singers. The ones who hold the microphone six inches or more from their mouths and get softer when they hear themselves. Experience will eventually help with part of this issue, but trusting the sound person is another important factor.
When a singer can be confident any sour notes they hit are not going to overshadow everything else, they tend to relax and sing more fully.
This requires attentiveness on the part of the audio engineer. Pay attention during rehearsals to who may not be as prepared as they should be or is just struggling with a particular part that week . Know if and when you may need to adjust their volume down. My mixing style tends to keep vocals close to the music anyway, but I try to be aware of spots where I may need to bury something for a second or so.
Obviously you can’t do this as much with a lead singer as you can background singers, and there is only so much overall we can do. But knowing you are going to “protect” them and not hang them out to dry, will help them sing more confidently. This reassurance from you, as the one in control of the sound, should always be accompanied with proper mic technique instructions and a brief lesson on the proximity effect.
The same goes for musicians and speakers as well. I’ve had to pull instruments out of the mix more than once because their playing was off. If the team knows you care and will handle those awkward situations to minimize the effect, they will become more comfortable quicker.