No DNI needed. The easiest way to join is via web. Login to your account via Mi Claro (prepaid is fine) and then once you are on the account page click on the "ClaroClub" button. Once you are in the club you can see all the offers incl. the 2x1 for Cinemark etc. and get it via web.
No Mi Claro is just the account management interface. I joined the ClaroClub via Mi Claro without problems. According to their website you can also join by sending an sms to 2477 with the word ClaroClub: "Por SMS: Podés adherirte a Claro Club enviando «ClaroClub» al 2477”.". But I have never tried it via SMS.
I was on prepaid when I joined the "ClaroClub" but had upgraded to a plan later on . So I was upgraded to ClaroClub "Red" (the only difference I can see to the prepaid level is higher discounts when buying a new phone).
Here is the page with the different club levels they have: