Mumbai feels completely insane when you first arrive, and that feeling may never go away. From a traveler standpoint it’s definitely a city of contrasts. With over 20 million people in the area it’s surprising that the tourist infrastructure still isn’t well built out yet. This is a major business travel hub, and with space at such a premium in the city center it means that budget hotels are hard to find, and also far more expensive than you’d expect. The pleasant southern neighborhood of Colaba is where you’ll find most non-business hotels, and most of the cheaper hotels even in that neighborhood feel like they are falling apart.
If you are coming to India then you should definitely spend a day or two or three in Mumbai, but probably not any more than that. With the exception of Delhi, which suffers from a similar situation, you can find hotels starting well under US$10 that are nice, and rooms in houses as low as US$1. Plus the real magic of India can not be found in its crowded mega-cities.
Good news about Mumbai is, except for hotels, most things here are quite cheap. Meals on the street can be found for practically pennies, and good local restaurants start just a bit more than that. Many of the most interesting things to see in the city are free as well so really once you find a hotel you can afford then everything else is very affordable.