Well ideally adding a class to the body would break the encapsulation that React components provide and fiddling with the DOM outside React could cause trouble if the body gets re-rendered. If possible, rather than adding the class to the document body, I would just add it to a component root element that React manages.
But to answer your question, you could do that way, but how often is your this.state.touchMode would change? If it's something that only changes during mount/unmount of the component, you can do it in componentWillMount (so that it only runs once when component mount, rather than every single time during render):
componentWillMount: function(){
document.body.classList.add('touchMode');
},
componentWillUnmount: function(){
document.body.classList.remove('touchMode');
}