My itinerary requires me to switch from the international to the the domestic airport at Mumbai.
I would like to know if I can exit the international terminal to meet my brother and then enter the domestic airport and continue my flight?
Yes, you can leave the international airport once you clear immigration and customs; meet your brother, then go to the domestic terminal.
As a visitor to India, you will need to have either a visa or are exempt from a visa (for example, you are an Indian citizen). Therefore, you can choose to enter the country at the international airport.
Here are the things you need to worry about:
Confirm with the airline if you need to collect your baggage at the international airport for customs inspection or not.
Make sure you have a boarding pass for your next flight, otherwise you will not be allowed through the security area of the domestic terminal. If you don't have one it is best to collect it from the transfer desk before you exit the international terminal.
It is up to you to make sure you don't miss your flight; as you have exited the transit area. The airline may choose to remove itself of any liability; as connection times assume the passengers are going to be within the transit zone.