Thats a hard question to answer, actually.
This setting basically means everyone can use your API without using authentication mechanics defined by activitypub. It does not mean all information is publicly available. To make it clear, lets have an example:
You register on some Funkwhale instance, create a private library and upload your music there. Next to it you create a podcast channel and publish your content.
If the "API requires authentication" flag is set, everyone on other instances is able to listen to your podcast. Your library is still private and you can grant access to it to other users of Funkwhale. Everyone else wont see any content on your pods page, since they do not authenticate.
If you disable the option, your podcast will be available to everyone. Your library is still private and you can still grant access to it to other Funkwhale users.
If this is dangerous for you or not, is something you need to evaluate. It highly depends where you live, how the laws are and how you apply moderation to your instance.
Take care and enjoy!