If you are asking so that you can create a policy for your center/gym it is probably helpful to see what the policies are at a few of your local facilities that are similar to your own.
Paul points out that responsibility and training may be varied across a spectrum of ages. Some children learn early either through parents who have home equipment, or because they are involved in sports programs. Some adults have never used equipment and do nut understand either gym etiquette or safe form. This is why any person first entering a facility, regardless of age needs to be offered orientation.
However, for legal reasons a gym typically cannot adopt a ‘well, you seem to know what you are doing, but you cannot come in’ policy. So most set guidelines that include age limits. They also typically offer structured training programs for youth, who are the most likely not to have been exposed to the gym setting. I often have seen those guidelines written with tiers: certain parts of the facility become accessible as certain ages are passed.
There is also the fact that certain types/amounts of exercise are not appropriate for youth until the bones have fused. This is kind of like with learning wind instruments where they want to wait until the teeth and jaw are ready. And if you allow kids under a certain age into the free weight room often they will want to try the weights, even if they have been told they should wait. It is partly attentionality, and partly maturity, and partly self control.
At our community Rec Center, you have to be 16 & up to be in the exercise area unsupervised. 13 – 15 can be in the exercise area under the direct supervision of an adult. I also offer a 4 hour (over 4 days) Youth Equipment Orientation class, which we cover the use of all machines – cardio/strength etc. as well as basic principles of fitness (overload/FITT, etc.) Once they complete the class, those 13 – 15 year olds are allowed upstairs in the fitness area but they must wear a bracelet indicating that they have passed the orientation class.
Hope that helps!