I recently bought my first property in West Warwick and had some conversations with the building official about the "apartment/in-law" in the basement. Which you may have guessed is not legal by my use of the quotations.
It is grandfathered in as a two-family but from talking with the building inspector there would need to be more lot square footage as well as enough parking spots to make it a three family. I assume Providence also has similar requirements although I can't be sure.
One option would be to try and finish it as part of one of the units. Of course this would still require all the codes for egress and everything be met, but you could do it without a zoning change and charge higher rent for the larger apartment. Another option (although I'm not sure about legal requirements for this) would be to make some storage units in the basement if it has good access. Possibly much better ROI because it would cost a lot less to put in.
I'm currently trying to figure out how to make the basement area legally part of one of the apartments. Which would turn the unit from about 1050 square feet, 2bed 1 bath to something like 1500+ square feet 3 bed 2 bath. Good luck.