With millions of people and a huge slum population, you might think Mumbai is dangerous for expats. You'll be pleased to know that it isn't - Mumbai is a generally safe city, so long as you are cautious.
Your biggest risks in Mumbai are the bugs and the traffic. Protect yourself from mosquitoes by wearing repellent such as "Odomos." Be very careful walking on sidewalks and in the streets. Drivers do not stop for pedestrians, so you will have to learn how to dart across streets quickly when there is a gap in the traffic.
Mumbai, like any other big city, has problems with petty theft. Be street smart - don't flash your money (or carry wads of it), store your wallet in your front pocket, and keep a good grip on your purse. Be as cautious as you would be in New York City or London.
There are occasional cases of rape in Mumbai, but risk of rape is not a cause for major concern. Women should, however, dress modestly. Keep shoulders covered and don't wear revealing or tight-fitting clothing. Dressing in loose, modest dress will keep away the stares and keep you cool in the Mumbai heat. It is safer to go outside with a friend or partner at night than it is to go alone.