On the geocaching.com site go to settings > authorisations then tick the box that says:
Do not allow Authorized Developer applications to access my public profile information. Learn more
I dont think it stops everything though. I daresay through the T's and C's you agree to via your geocaching.com account you agree to share certain info via API.
edit: Only way to not share logs is to delete them, as per here: https://www.geocaching.com/help/index.php?pg=kb.chapter&id=27&pgid=920