Since Babel 6 use babel-register hook to make on-the-fly transpilation.

First:

 npm install babel-register

Then require it with:

require('babel-register');    
// not using 
// require('babel/register');
// or 
// require('babel-core/register);

To Convert your Ecmascript 6 code to ecmascript 5, you must set Babel presets option with require babel-register Like this:

require('babel-register')({
  presets: [ 'es2015' ]
});

Unlike the answer of @alexander-pustovalov you do not need to .babelrc file.

you must also install babel-preset-es2015:

npm install babel-preset-es2015

Finally your Server.js file will be:

require('babel-register')({
   presets: [ 'es2015' ]
});

const env = process.env.NODE_ENV || 'development';
const port = process.env.NODE_PORT || 1995;

const http = require('http');
const express = require('express');
const address = require('network-address');

let app = express();

app.set('port', port);
app.use(express.static(path.join(__dirname, 'public')));

app.get('*', (req, res) => {
   res.send('Hello!');
});

http.createServer(app).listen(app.get('port'), function () {
   console.info('Demo app is listening on "%s:%s" env="%s"', address(), app.get('port'), env);
});

require('babel/register') doesn't transpile the file it is called from. If you want server.js to be included in on-the-fly transpilation, you should execute it with babel-node (Babel's CLI replacement for node).

See my answer here for an example.

I ran into a similar issue trying to render a react page (.jsx) on the server. I fixed it by putting the snippet below at the top of my server file

require('babel-register')({
    presets: ['es2015', 'react']
});

make sure you have npm babel-preset-es2015 and babel-preset-react installed