I am a Disney newbie but Nassau veteran, regularly traveling there since the 1980's. The crime has gotten bad, compared to better days. I read the Nassau Guardian and am surprised by the crime the island is now compared to the past. I would not walk around downtown as a mother of young kids myself these days, particularly since you have never been.
But there are still great things to do. For a really low-key, laid back excursion with your little ones, the Adastra Gardens is amazing. It isn't Disney, as it is very laid back, but the scale of it and the interactions are perfect for your kids.
The flamingo show is fun, and there are so many great photo ops with the kids and the different birds and animals.
I would just take a taxi there and have the gardens call you one when you are ready to leave.