Crowded at any time of day, the increasing number of cars in Mumbai has done nothing to alleviate the crowded public transport. The famous red double-decker buses that connect everywhere in the city are usually overcrowded, and strangled by the traffic, meaning that it can take longer to use transport than to walk from place to place.
There are swarms yellow and black taxis everywhere in the city, and these fare better than the buses in terms of getting about - usually because the drivers are unconcerned about safety, and will aim for any gap in the traffic no matter how small it may look.
If you're prepared to put up with the frightening congestion and frustration of sitting in queues all day, then renting a car in Mumbai is a real option, although in truth, its only worth doing if you're planning to explore the surrounding areas.