How to get into the city from the Mumbai airport (BOM)
When I visited earlier this year, taxi was pretty much the only way to get to downtown from the Mumbai airport. I checked for bus and train options, but neither of these were good options (e.g. there's no train station at the airport, I had to take a bus/taxi to the closest train station)
Fortunately, taxi was really cheap in Mumbai. I took one of the prepaid taxis from the airport to my hotel downtown (Fort area) and I paid 500 rupees (~$8 US). Trip took about 1 hour
There are 2 kinds of taxis at the airport: prepaid and metered. Go with prepaid. Metered taxis from what I understand always come out to be 20-30% more expensive than prepaid, which is fixed price
There are many of these taxi counters at the arrival area after you exit the luggage pickup area with signs in English. You won't miss them
For both prepaid and metered taxis, make sure you have enough cash on hand. They don't take credit card. Exchange about $20-$40 USD at the airport is what I recommend
These taxi counters operate 24/7, so you don't have to worry about your arrival time
Alternatively, you can try Uber (or Ola, which is Indian version of Uber). They come out to 10-20% cheaper than the prepaid taxis, and obviously you pay by credit card which is nice. To get an Uber/Ola: go to one of the info counters at the arrival area and ask for a wifi code to get on the wifi network. Then use your Uber/Ola app to get a car. Alternatively you can get an Indian SIM card at the airport.