There's a misconception here. The word der has various roles. In the film title
Das Leben der Anderen (genitive)
it is not the definite masculine article der, but the genitive inflection of the definite plural article die (die Anderen is plural). A proper translation would be
The life of the Others
(Caution: It differs from the English film title The Lives of Others, which would be Die Leben anderer in German!)
where the German article der is reflected directly by the English article the. Notice, that in English the preposition of is used, which actually corresponds to the dative preposition von in German. In fact, the film title could be reworded with a dative instead of a genitive:
Das Leben von den Anderen (dative)
But this sounds clumsy, which is why I think the author preferred the genitive version. I guess, this is different in English, where the genitive version "The Others’ life" would sound clumsy.