React.js files syntax in sublime text 3

Just installing it isn't enough, you have to also tell SublimeText to use it.

Either do

a) Ctrl-Shift-P, type "Babel" and select Set Syntax: Javascript(Babel)

or

b) Go to the menu and do View->Syntax->Open all with current extension as...->Babel->Javascript(Babel).

You could follow the below mentioned steps:-

Setting babel as the default syntax

To set it as the default syntax for a particular extension:

1. Open a file with that extension,
2. Select View from the menu,
3. Then Syntax -> Open all with current extension as... -> Babel -> JavaScript (Babel).
4. Repeat this for each extension (e.g.: .js and .jsx).

Setting a Color Scheme

Babel comes bundled with Next and Monokai from Benvie/JavaScriptNext.tmLanguage. Select one from Preferences -> Color Scheme -> Babel

Courtesy: https://github.com/babel/babel-sublime

This might possibly help you out...follow below steps

1. press Ctrl + Shift + P
2. Doing so, Command Palette pops up and there type : Package control . And then select the option Package Control: install package
3. Now typein and select 'Babel'
4. Now open 'View' menu and Navigate to Syntax, Open all with current extension as…, Babel, Javascript (Babel).