Node.js REST API versioning the right way?

I would recommend using node-express-versioning module instead.

It would help you to support multiple versions without changing the url of API, just send the version of API and direct the call to that version route-controller.

*

*//version taken out from header

    app.use(function(req, res, next)
    {
       req.version = req.headers['accept-version'];
       console.log(req.version);
       next();
    });

    //version path defined

    app.use('/api', versionRoutes({  
       "1.0.0": respondV1,
       "2.0.0": respondV2
    }));

    function respondV1(req, res, next)
     {   
        app.use('/api',routeV1);
        next();
     }
    function respondV2(req, res, next)
    {
       app.use('/api',routeV2);
       next();
    }*

*

Instead of adding version in every route you can add it in app level. So It won't be tightly coupled with API route.

import * as express from 'express';

// v1/get-ride.js
const router = express.Router();
router.post('/rides/:id', dep.verifyToken(), (req, res) => {
    // Your code
});
app.use('/v1', router);


// v2/get-ride.js
const router = express.Router();
router.post('/rides/:id', dep.verifyToken(), (req, res) => {
    // Your code
});
app.use('/v2', router);

There's no "right way" to do API versioning.

However, URI based global versioning is not at all RESTful.

Phil Sturgeon of "APIs you won't hate" recommends an API Evolution approach instead.

See: https://apisyouwonthate.com/blog/api-evolution-for-rest-http-apis

Have you considered this as an option?