Coins were originally the only "hard" currency. For example, in England, circa 1780, a tradesman might earn 20 pounds a year. That could be paid, easily, in coins: guineas, shillings, pence. At this time banks could actually issue their own paper money ( yes, that's where we get the term bank notes) . Because there was only so much gold , "specie" currency was often in short supply, and bank notes were not uncommon. But of course they depended on the security of that bank, and in the mid 1800's the Bank of England was given the sole right to print money.

Much of what you're seeing is just the consequences of inflation over a very long time. In 1850, a laborer might earn 50 cent a day. In 1900, that might have gone up to $1 a day: Henry Ford made a real impact when he first offered $5 at his auto plant. A good lunch in 1925 in New York might cost you 50 cents, and you might leave a nickel as a tip. There was a counterfeit ring in the 1930's in West Virginia that was content to make quarters. A small house in a midwestern US town in 1930 might cost several hundred dollars. Paper money and inflation also became more dominant as currencies stopped being based on gold or silver. But the history of that, and the Gold Standard, is a much bigger economic topic.

There have been times when it's found to be too expensive to mint coins- when the face value of the coin is less than the cost of the metal and the production. In the later 1700's, the English government became very slow to mint small coins. As the economy was heating up with the Industrial Revolution, this resulted in shortages of them. The result was a proliferation of merchants and businessmen making small token coins to be used for small transactions. These have become known as Conder Tokens, named for a collector of them. (The link is merely to the Wikipedia article, but it's actually a rather nice introduction to them...at least, it is today. You can also see more pictures of some here ) . When minting began to be done on an industrial basis, this shortage became less troublesome.

However, the cost of producing vs value of coins has often been a question of balance. Though the long life of a dollar coin makes it theoretically better, compared to the short life of a dollar bill, there have been proposals for some years to stop minting pennies in the US, which cost more to make than they are worth, and which tend to disappear ( end up in small piles in someone's drawer, or in a jar) rather than circulate.