Take Walmart as an example. It is the world's largest employer. It's wage structure is such that many employees are encouraged to apply for federal assistance. So, society is paying part of the cost of what should be proper wages. And the workers have no surplus funds to feed back into the business community. Their lack of spending power translates into health and educational issues - even criminal issues - which further burden government.
If we take a casual tour of business operations, as the average citizen may see it, we might get the following picture of what would happen if minimum wages were arbitrarily raised. From the business owner's viewpoint, he would have to accept less profits or, if his business operated on a very slim margin he would have to raise prices of his product or services with the following consequences.
Theoretically, in a highly competitive market that necessitated the slim margin, higher prices may not be feasible for all competitors, so the less efficient businesses would either lay-off some staff - thus increasing their work burden, reduce other costs by slashing quality, or go out or business altogether. If all competitors were forced to accept price increases, then the market demand for the product or services might decrease according to how essential they are to the consumer.
If the burden on the government of sub-living wages were compensated by taxing businesses, then business might see the wisdom of raising the minimum wage as an alternative. The thing to keep in mind is that it is the consumer - not the business - that ultimately bears the burden of raising minimum wages. Also, it is wages that fuel the economy by purchasing power.
In a more advanced and efficiently-run society without the burden of wars and other destructive economic ailments, there could be entirely different criteria for determining wages that would result in greater equality and fulfillment of life. So government itself could be the greatest impediment to better living standards.