What is the primary responsibility of an anthropologist?

To study human culture. Anthropologists must strive, in their work, to set aside their own culture when studying another culture (or studying their own culture) to avoid ethnocentrism (a bias toward one's own culture). Avoiding ethnocentrism is difficult for any anthropologist the first time they are in the field.

Anthropologists tend to strive to prevent ethnicide, the destruction of cultures (not to be confused with ethnic genocide, which is the mass slaughter of the people of a particular culture); this means trying to protect waning indigenous tribes from losing their culture by being absorbed into their developing nature, exploited by first-world corporations, or converted by missionaries ... or, if unable to act to avoid the ethnicide, to at least record in every possible detail the culture being lost.