If you are writing your React code with JSX and ES6 syntax you must transpile at least the client code with babel. You have two ways of doing this:
The disadvantage of transpiling only the client side code is that you need to regenerate the bundle on every code change while developing which makes development slow.
I write more about this in my case study of SSR of my current project that you can read here blog.jakoblind.no/case-study-ssr-react