Pre-paid taxis are the recommended way of reaching TIFR from Mumbai's airports. Specify your destination as "Navy Nagar, Colaba". You will be given a receipt for your payment. Retain this till the end of your ride.
From Chhatrapati Shivaji Terminus (also known as Victoria Terminus) Railway Station
Taxis are available from the Chhatrapati Shivaji Terminus. They charged by meter. Alternatively, buses (routes 3, 11, 125) come to Navy Nagar from outside CST. TIFR is on the same road as the Navy Nagar bus terminus (last stop), which is where you should get off.
From Mumbai Central Railway station
Taxis are available from the ranks outside the station. Alternatively, you could cross over to the local station and board a Churchgate bound local. You have to buy a separate ticket for the local train. Taxis are available from outside Churchgate. Also, a bus (route 137) comes from Churchgate to the Navy Nagar terminus, down the road from TIFR.
From Lokmanya Tilak Terminus Railway Station
Prepaid taxis are the best option. Prepaid Taxi counter is available just outside platform number 1.