Amarsons Park in Breach Candy is dotted with cute animal pictures and random fantastical elements (see mermaids below), keeping kids entertained for hours. The park charges Rs. 10 as an entrance fee and therefore keeps out stragglers and pervs, and is by the sea which is always a pleasant site in Bombay.
Band Stand in Churchgate had a massive overhaul about 10 years ago and is a very sizable park with many jungle gyms, swings, and climbing options. The highlight is the horse and pony rides offered at the very entrance of the park. This being a very public park, it is critical that you keep your child in your sight at all times and never let him/her talk to strangers.
Other posts in the Mumbai series:
India - General Travel Tips
Bombay (Mumbai), India - Most Baby-Friendly Club