"It seems to me to be too dangerous to allow inexperienced people - especially children - that close to those [horned] animals."
We never allow inexperienced people "in with" cattle, unless accompanied by a farm hand. There are several places on the farm where inexperienced people can interact with our animals in a controlled environment; where the animals are behind a wall (that they can stick their head through), or are behind electric fences. Additionally, we ask anyone taunting or cajoling the animals to leave the farm immediately because they could be enticing the animals to misbehave.
Our animals are bred for docility and temperment. Don't forget, we have to handle them too! Furthermore, any animal exhibiting aggressive behavior is sent to slaughter as quickly as possible, and non-farm workers are not permitted near an aggressive animal until it is removed from the farm. Also, a lot of people have a pronounced fear of horned animals. To put this in perspective, there are far more traffic deaths than gorings caused by horned animals. See http://www.cdc.gov/Mmwr/preview/mmwrhtml/mm5829a2.htm. Also, most cattle-related injuries/deaths are not caused by horns, but by blunt force injury to the chest and/or head. So the animal's weight/size difference tends to be the determining factor in injuries/deaths more than horns. In spite of all the above, we request that visitors please keep in mind that cattle can have more than ten times the weight of an adult human, and to keep this in mind with interacting with the animals. Do not place your hands or other body parts between an animal and an immovable object!
During the "cow riding" events, there is an experienced handler with the animal, and someone (usually the parent) with the rider. The animals walk slowly and are not permitted to walk freely with a rider. Only certain qualified and trained animals are permitted to work the cow riding events.
Because the animals can toss their heads (typically to ward off bugs or stray hair), we will frequently "ball" the tips of their horns during events to prevent accidental injury to handlers and riders. However, the balls are removed between events to permit the animals to use their horns in their natural state. (There are predatory dog packs in the region, and they have taken one of our calves in the past.)