You can:
1) Kill the guard to the mine, drops a key.
2) Bribe the guard for the key.
3) Pressure the guard if your persuasion skill is high enough.
4) Finally, you can talk to the head guy in town (like the tribal leader) his name eludes me. He will give you a pass for the mine to help them out with the diseased queen. It lets you kill two birds with one stone.

Vox