First of all, remember that it's a privilege to serve in children's ministry. It's no one's "right" to serve in children's ministry. Yes, children's ministry is a place where people can use their gifts and talents, but that can never take precedent over the safety and security of the children. Never hesitate to say "no" to someone who wants to serve in children's ministry but whose background or life situation disqualifies them.
Make sure you run a background check on every single person that serves in your children's ministry. This should be done before they are cleared to start serving. Yes, it costs money, but you can't afford not to do this.
Personally interview everyone who wants to serve in your children's ministry. Sit down with them, look them in the eye and ask them the hard questions. Here are the questions we ask.
Never allow anyone to be alone with a child. There should always be two adults in every room. Don't open the room until they are in place. If a child has to be walked to the bathroom, make sure two adults walk him or her there.
Avoid off-campus events at houses. Sleepovers at a volunteer's house, small group parties and other events outside the walls of the church increase the risk of abuse happening. If you do allow these, I would strongly encourage a staff member to be present and still avoid overnight events in homes.
Make sure all your classrooms have windows so you can see inside. Avoid situations where there are closed doors with no view inside the classroom.
If you'd like more tips and information, here are some more helpful articles.
A Way to See If the Kids In Your Ministry Are Safe
Parents Expectations About Safety and Church and How to Meet Them
The Penn State Abuse Tragedy...How to Keep It From Happening in Your Ministry
Children's Ministry Safety and Security Podcast