I'm surprised that only one person pointed out that the traffic in Mumbai is always so bad that, unless you have a very large amount of schedule flexibility (or "free" time), you really need to stay near where you need to be in Mumbai (and Delhi; Bangalore is a little more tractable, as is Pune).
My business takes me twice every year to three north central Indian cities, and in Mumbai and Delhi (Gurgaon) I always stay near my work sites to save time stuck in traffic.
In Mumbai, my work is in the suburb of Powai, so I stay at the Renaissance Mumbai Hotel & Convention Centre (
http://marriott.com/property/propertypage/BOMBR). The hotel is reliable, clean, and has good service. For example, they once sent out a car to pick up a refill card for my India based mobile phone for me so that it would be waiting for me on my arrival. The food is good but not spectacular, including the breakfast buffet which is usually included in the room price. The location on the lake is nice, and importantly it's quiet. The hotel has a full featured gym (a dozen or so treadmills, bicycles &etc, a full set of weight machines, an aerobics / stretching floor, and more) which is open 24 hours a day and is included in the room price. The hotel has always given me at least a one way pick up from the airport in whatever room rate I picked, and sometimes has thrown in the return to the airport as well. My room rate at the Renaissance last month was US$169/night including airport transfers both ways, and breakfast each morning.
What I haven't been offered at any major Indian hotel yet is free Internet access. They get me for $10 to $20 per day for that, but it's critical for me to have it so I (my employer) pay(s). Ouch.
By the way, I second the recommendation of using the hotel car service for airport pickup if you haven't been to India before. Flights to India almost all land between 11:00pm and 1:00am so that they can leave and return to points elsewhere in the world at more convenient times for those other places. (The exception is a very small number of evening departures from London which land at nice comfortable morning hours in India, but if you aren't originating in London, those mean a long wait or a hotel stay between flight legs). As the other poster said, work the hotel car airport pickup in to the room rate.
Once you get comfortable with the late hour and somewhat chaotic situation of an airport arrival in Mumbai or Delhi, I believe that the airport prepaid taxis are available in air conditioned versions. Just ask to be sure.
The other hotel I've stayed at in the past in Mumbai was the ITC Sheraton which was quite excellent, if that's the part of the city you need to be in. I've never stayed downtown where, as others have said, most of the tourist stuff is located.
Finally, though the hotel cars are good for getting to the hotel from the airport, I recommend against renting your car (& driver) from the hotel for your daily needs - instead, call for a "cool cab" (air conditioned taxi). The hotels add 30% to 70% on top of the actual rates for cars, and while the hotel cars will be newer and nicer, almost any cool cab will be adequate to the task of getting you around the city, and the hotel car drivers do not necessarily speak better English than the cool cab services' drivers. Bring your intended destination address(es) written on paper in any case.
Enjoy India! Yes, it can be crowded and chaotic, and at times its weather is horrific. It is also full of really nice people, a stunning mix of cultures and colors, really good food, and is largely very affordable if your home economy is in US/Canada/Europe/Japan/&etc.
Cheers!
-Jay in Atlanta