Does the Diplomacy/Trading skill have any affect on mining?

“A character can use this skill to increase public opinion for their group, and maximize profits and reduce losses of their business. ”

Would it affect the cost associated with extraction?

Nope, that part with reducing losses is for negative FI. As for mining and cost:

Cost = Yield * RM price + (Yield * RM price * Tax) + (0.2 * Yield * RM price * Crime) + 0.375 * NPC_Cost * MAX(0, NPC-1) / (Management+1)

As you can see, management skill is one that have influence on mining. Unless something changed of lately and I've missed it.