As a VERY strong advocate for Smoke "Alarms" and AFCI's , I am 100% sure I should not make this statement.....I do not see "basements" listed in Section 210.12(A) as directed back to in Section 210.12(B), unless you are going to reach and use the "similar rooms or areas" aspect of the code reference.
If wish to call it a "recreation room" then yes.....you are modifying the branch circuit so AFCI protection is required. If you are calling it a "Family Room" then yes...you are modifying the branch circuit and AFCI protection is required.
Would I install AFCI Protection regardless...YES...because I do drink from the Kool-Aid that says they work and add benefit.