Your first response might be marketing or customer support. But I think the question really needs to focus on what is your goal with social media. If you goal is brand management then marketing might make sense. If you goal is to decrease customer support then your support team, maybe. What if it was a collective effort versus a silo approach? Where each group had folks not dedicated, but individuals interested in connect with the community and sharing their knowledge.
Now that would be interesting. Could that actually occur in an organization or would it driven by one department fighting for control?