I like the "Atlantic" side beach, ie the south-facing one. There's plenty of free parking over the road from the beach, it's not crowded, and Key West is sufficiently small that you can drive there from your accommodation in a few minutes.

Recent times I've used the Southernmost at the quiet end of Duval Street www.oldtownresorts.com , because you can walk up into town in a few minutes - and that is a big plus in Key West, compared to staying outside town on the main highway, driving in, and not finding any parking. Should suit the family.

Hope you are driving there from Miami - the Overseas Highway is one of my favourite drives anywhere.