It is very, very unlikely that a genuine child would be able to play a sport with enough skill to be in the Olympics. But if somehow there was a child who had enough athletic prowess to go head to head with the world's top adults and teenagers, they should absolutely be allowed to compete.
Concerns that the hypothetical child would be pushing themself too hard or that their athletic training interferes with their growth, health, safety, schoolwork, emotional health, etc. are absolutely valid. But a child who is training at an international elite level would already be putting those things aside just to play/do their sport. You might as well ban children training for sports at an international level in the first place, which will never happen (although that is a whole other argument).
Certainly, there would have to be strict regulations, but a kid that talented would already be training at a ridiculous and taxing level.
There are plenty of Olympic sports appropriate for children to play when they are young. Among them are basketball, field hockey, badminton, table tennis, soccer, golf (new in 2016), swimming, and more. Some sports are too dangerous for young kids like luge, ski jumping, bobsledding, white water rafting and the like. In general, getting kids some physical activity such as an Olympic sport is a great idea.
The Olympics are for adults, and it should stay that way. Children have enough pressure from society as it is, and they don't need anything else forcing them to grow up before their time. Olympic sports require hard training and lots of dedication, and it would be wrong to subject a child to that type of schedule. Childhood is a time for children to play and learn. Olympic sports would just complicate things for them, and that is possibly why they are geared towards the older crowd anyway.