Both airports are in the western suburbs of northern Mumbai. If you're staying in South Mumbai, it will be a one-hour taxi, bus, or car ride into town. Juhu, where a lot of other hotels are located, is about 20 minutes away.
If you arrive by train, chances are that you will show up at the Chhatrapati Shivaji Terminus, in south Mumbai, or at Bombay Central, in central Mumbai. Chhatrapati Shivaji Terminus is a few minutes away from most hotels in South Mumbai, but an hour away from Juhu. From Bombay Central, south Mumbai is 20 minutes away and Juhu and north Mumbai about 40 minutes. A few trains arrive in the northern suburb of Bandra, which is closer to Juhu and the airport. A few others arrive at the very chaotic and oddly located Kurla Terminus—from there both Juhu and south Mumbai are about an hour by taxi.
Mumbai's sights are spread out, so getting around by taxi is a sensible option. You can flag down yellow-top black taxis or silver-and-blue air-conditioned taxis anywhere in the city. Insist that the driver turn on the meter, a rusty contraption on the hood of the car, before setting off. Because the meters haven't been adjusted for quite some time, it takes some arithmetic to compute the latest (higher) fares, based on the meter reading. Drivers must show you their revised tariff cards, but they sometimes "misplace" them, or whip out a chart for air-conditioned cabs, or show you fares chargeable after midnight. At this writing the legal fare was 14 times the total amount shown on the meter, based on roughly Rs. 13 for the first kilometer and about Rs. 1.5 for each additional kilometer. Air-conditioned taxi fares are 25% higher—the starting rate is Rs. 16.50 and Rs. 12 for each additional kilometer. If you are short of time you might consider hiring a car and driver.
Auto-rickshaws are permitted only in Mumbai's suburbs, beyond Bandra, where you can flag them down on the street. As with regular taxis, insist on paying by the meter and ask to see the tariff card. Mumbai has a good bus system; however, navigating long routes and big crowds is likely to turn trips into an uphill battle.