Why doesn't the price of gold drop as mining occurs that produces more gold? According to my common knowledge, the price should drop when supply increases. Why doesn't the price of gold drop slowly as more gold is produced from mining?
You cannot reason directly from a price change. Prices may increase because supply has fallen or demand has changed. In general we can't distinguish between the two unless we know something about the demand and supply curve in particular and not just prices and quantities. 

For example, when demand increases (the demand curve shifts to the right but supply curves are held fixed) this increases demand for gold at all prices. Which means that equilibrium quantity increases (more gold is produced) as well.  

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But that gold doesn't decrease prices. The only reason more gold was produced was because prices were higher. If prices fell in response to higher quantity produced the producers (who have the same supply function by assumption) would not produce more. So in response to a demand shock for gold prices increase in order to clear the market for gold. Otherwise there is not enough gold at the old prices.