A key aspect of a basement apartment are the entrances/exits. Many people want to finish their basement only to find out that it will be more work than they had hoped because they don't have the required access.
If you're finishing your basement for a family member or someone you're close to and you don't mind sharing the rest of the house, you'll most likely have a door that connects the basement to the rest of the house. If your basement is below ground, you'll probably also need an egress window for a second exit point. An egress window has to be a certain size determined by your local codes and be big enough for people to escape through in the event of an emergency. If you have a bedroom in your basement, it must have egress. If you are finishing more of an open studio space, you should check your local codes.
If you are finishing your basement as a separate apartment and trying to maximize rental income, you'll need a separate entrance for the people living there. It's possible to build an entrance, but it will mean cutting into your foundation and digging out a landing and space for stairs. Whether or not your basement has an exterior entrance is a big factor for people trying to figure out their options.
You'll also need to examine how the basement is currently connected to your main house. If there is a doorway between the two, you'll need to look at closing it off, or complying with local building codes, which might mean adding a fire door to create fire separation between the two units.
Lots to think about, but the good thing about basements is that your options are limited and so decisions are easier.