So what's a safe age for kids to start running as a sport, rather than just for fun? Three years old is a little young for kids to start a formal running program. They may not "get" the concept of running a race and it could be a miserable experience for everyone.
But if your child shows an interest in running, kindergarten is a good time to look for a youth running program or enter your child in a local kids' race (usually short distances of 100 – 400 meters). But make sure that the program isn't too regimented or intense. The idea is for kids to get some exercise, have fun, and learn to love running.
If your child shows an interest in running before the age of 5, you definitely don't want to discourage him or her. You can encourage her to run by playing tag, doing an obstacle course, even chasing after the dog -- as long as it doesn't feel like a formal, structured program.