The best time to visit Mumbai is the winter season. If you want to know months, it is from October to February. The weather remains pleasant. Mumbai doesn't become much colder in winters and temperature hardly goes down to 15 degree C. So, you can explore different places in or around the city.
Avoid visit Mumbai during the monsoon because it receives heavy rain fall and it becomes really hard to explore the city. The showers are also unpredictable. Summer is also not good because Mumbai becomes too hot and humid.