• “had” is a preterit, which is a past tense.

• In French you say “avoir le potentiel de faire quelque chose”

• “faire” is very vague in French, you should avoid using it as a translation from “make”. “faire” is closer to “do”. Here, use “fabriquer” for example.

• “duquel” doesn’t mean that. Here some examples of the correct usage: http://www.wordreference.com/fren/duquel. “Quel” is not correct either. “De ce qui/de ce que” is definitely what you need here.

• “could” is a conditional, translate it by “pourrait”.

• I wonder if even the English phrase is correct here by using “possible”. I think they mean “possibly” in which case your translation (“possiblement”) is good.

• “l’existence” is feminine in French, so it is “de l’existence”.

• “le cosmos” is singular.

Ce qui, donne, concaténé:

Cette génération a le potentiel de poser les fondations de ce que
pourrait être le futur de l’existence humaine dans le cosmos.

C’est un peu lourd, en inversant le verbe de la seconde partie:

Cette génération a le potentiel de poser les fondations de ce que
le futur de l’existence humaine dans le cosmos pourrait être.