There are direct buses that ply from Mumbai and Thane to Murud Janjira. If you are driving down the best route is via Roha. Follow the NH17 till Roha from where you can take the SH connecting Roha to Murud.
There are direct M.S.R.T.C buses that ply from Swargate to Murud Janjira. If you are driving down the shortest route is via Indapur. From Chandni Chowk take the Tamhini ghat route to reach Indapur. The alternative route is via Kolad to reach Roha. Follow the SH to Murud Janjira after that. The nearest railway station from Murud Janjira is Roha