Reactjs Tab Component
As there's not all that much code here to review, I've reviewed some style points:

You have some strange use of whitespace throughout your code:


handleOnClick(key, event) {

    event.preventDefault();

    this.setState({
        activeIndex : key
                // ^-- whitespace shouldn't be before a property colon
    });
}



and not necessarily that it's wrong, but there's a lot of empty whitespace lines that make your program look a lot beefier than it should.


child.props.active === true ? true : (active == index++)



You don't need to compare properties to booleans as simply specifying the variable without comparison performs a boolean comparison:

var thing = true;
console.log(thing === true ? 1 : 2); // identical
console.log(thing ? 1 : 2);          // identical


You've also got some inconsistency in your use of semicolons:


constructor() {

    super();

    this.state = {
        activeIndex : 0
    }
}