You could buy your packaged-for-tourists spices at hassle-y Crawford Market; or, you could go where Mumbaikars go, which is this fragrant market in Dadar that's considered top-rate for fresh, unadulterated hand-ground powdered goodness, fresh chillies and other chef essentials.
Stroll down Estate Ln – the section of the market devoted to spices – where you'll find all manner of spices, chillies, masalas, dal (lentils) and very little English. Ask around for Lakshmi General Stores. Normally, the spices are stored in clay pots known as matka (you can buy them down the lane at Jai Bharat) but they'll last six months if refrigerated at home otherwise.