In The Prize, author Daniel Yergin says that Chinese were drilling 3000-foot wells in their quest for salt for centuries before oil drilling began in Pennsylvania. And I know that in some places, pre-industrial societies obtained salt via evaporation. So mining salt in horrible conditions is not the only option.
But salt mines seem to have taken on an almost iconic status as a hellish work environment to which one would be sent as a punishment little short of death. Were they more dangerous than other mines (or other forms of penal labor)? Where, if anywhere, was being sent to work in salt mines really used as a form of punishment, and for what crimes?
A subreddit search turned up a few interesting things about salt, but nothing relevant.
Edit: word
You might be able to guess, working in closed spaces underground with little ventilation where you're breathing in particles of rock salt dehydrates you in the short term, and in the long term you're accidentally ingesting high levels of salt, and that increased sodium intake can have real effects on the body. It's very obviously difficult and grueling physical labor. However, it's not the salt mines themselves that created the reputation for being a nightmare.
I've heard stories of salt mines only being worked on by criminals and slaves during ancient Rome, but as far as it being used as a punishment, the most definite example I can find, and where the trope/idiom almost definitely comes from, is in 18th and 19th century Russia, when the government would send criminals to eastern Siberia to work in various mines. Some to gold mines, others for lead and silver, others salt mines, others to help build the Trans-Siberian Railroad etc. They would be brutally worked according to accounts, and punishments for not working hard enough were very severe. So it seems as though it's not just the salt mines specifically, but pretty much all Siberian labor camps for the way workers were treated by foremen.
19th century explorer George Kennan traveled quite a bit around Siberia/Eurasia and wrote about his travels. He published a book called Siberia and the Exile System in 1891 that had been released serially in a magazine. More importantly, before publishing this book, he spent a few years travelling around the US as a lecturer, telling people about the horrors of Siberian exile and his opposition to the Russian autocracy. His lectures were quite well attended, and it's estimated that he spoke to upwards of a million people on his circuit, before chapters of his book were released to an even wider audience.
According to the American Heritage Idiom Dictionary, the phrase "back to the salt mines" does indeed originate around the late 19th/early 20th century.