I don't think the aim here should be to culturally save just languages but the entire culture and lifestyle that spawns and nourishes them. At the very least, you'd have to put the brakes on the globalization movement, so as to preserve local cultural differentiation. It's very unlikely to happen.
A separate question is whether we have a duty to help _other_ cultures in their quest for self-preservation. I think there are altruistic reasons for doing so stemming from the fact that most people value their own culture so we can empathetically understand why others would want to preserve theirs.
There are also interesting biological and sociological arguments around diversity and resilience of the species that don't rely on altruism.
The question specifically about language is asking something different though. So this conversation should just be about languages where the group that spoke it no longer has enough interest to continue that tradition. So the question becomes should we save the language of other cultures for them? I don't think there is a strong case (other than academic preservation).
Arguments about the intrinsic relationship of language to culture are valid but fall apart when the members of the culture itself don't care enough to preserve the language.