Not costs, but benefits to the economy under a Gold Standard...
It does not. It creates a profit.
Of course, it does take effort to dig gold out of the ground. However, gold production never ceased after the end of the world gold standard in 1971. Roughly half of all the gold ever mined, in all of history, has been mined after 1971. Annual production today is the highest in history, and about double what it was in 1970. People seem happy to continue paying those "costs."
If one is to use gold coins, then someone needs to pay for this gold. Who pays? Is it "everyone"? The government? Taxpayers? Who?
Let's say an economy uses gold coins only. There is no paper money. (For simplicity, we will also assume no banks.) Someone works all week and gets a one-ounce gold coin in payment on Friday. The person has "paid" for this coin with a week's worth of work.
Now, let's have an economy with no gold coins, just paper banknotes linked to gold. Let's say there's a banknote worth one ounce of gold. (The US $20 banknote, before 1933, was worth about 0.97 of an ounce of gold.) Someone works all week, and gets a banknote worth one ounce of gold in payment on Friday.
The "cost" to the person of the gold coin and the banknote are the same. One week of work.
The government is in much the same situation. Whether its taxes are paid in coinage or in banknotes, the outcome is about the same.
The difference is at the currency issuer – the producer of the banknotes, which would be a central bank today. Let's say that the currency issuer has a "100% reserve" system. For every banknote, there is an equivalent amount of gold in a vault.
This situation is not much different than where gold coinage was used exclusively. The amount of gold is the same.
Now, let's say that the currency issuer has a 20% reserve, which was a typical level among private currency issuers in the US during the 19th century. The other 80% of reserves consists of interest-bearing debt. Today, that would most likely be government bonds.
The amount of gold used by the monetary system has now fallen by 80%, replaced by interest-bearing bonds. The interest income from the bonds produces a profit for the currency issuer.
If you had a floating fiat paper currency, with no gold at all, the "cost" of the money would still be the same – a week's worth of work. However, the assets of the currency issuer could be 100% interest-bearing debt.
Thus, we see that the "cost" of the money is the same in all cases – a week's worth of work. The question is the "profit." In a 100% bullion reserve system, there is no profit. However, with a 20% reserve system, there is quite a bit of profit. This profit accrues entirely to the currency issuer – today, a central bank.
We can also see that the profit enjoyed by the currency issuer doesn't really change much, from a 20% bullion reserve/80% debt gold standard system and a 100%-debt system. The difference is only the 20% portion. The other 80% is identical. (In practice, today's central banks still hold gold bullion reserves.) So, the gold standard system's profitability is actually much the same as the floating fiat system's profitability.
Some economists – including David Ricardo, in 1817 – have suggested ways of operating a gold standard system with no gold bullion reserves at all. The "gold exchange standards", or currency-board systems, common in the 20th century, were one example of this. They are "100% debt" systems.
I suggest that you shouldn't be too concerned about maximizing the profitability of central banks. They can look after it well enough themselves.
Concern yourself with the quality of the currency. For nearly two centuries, 1789 to 1971, the US embraced the principle of gold-based money, and became the world's economic superpower. Money was simple, stable, reliable and predictable. Despite short-term setbacks, the middle class grew steadily wealthier, generation after generation.
Today's economists talk a big talk, but in the past forty-five years of floating currencies, have they ever been able to produce that kind of result? Mostly, they just bounce from one crisis to another, blowing bubbles along the way and leaving a train of wreckage in their wake. Have they learned anything? They seem to have gotten pretty good at "kicking the can", avoiding a minor crisis with further distortions that lead to a bigger crisis later.
I think that there will eventually be a big enough crisis that people say: "Enough is enough. You've had your chance. Now it is time for you to go." But before then, we will have to know what we will replace them with.