I'm the author of the linked comment. I originally focused on state prisons because the article was about events in a state prison, so P(person did something bad|person is currently in state prison) was the relevant statistic. But let's repeat the analysis counting federal prisons and local jails as well.
In general, people with sentences longer than a year go to prison, and sentences shorter than a year go to jail. Roughly 57% of incarcerated people in the US are in state prisons, 9% in federal prisons, and 33% in jails [0]. In my earlier comment I noted that 52% of state prisoners are in for violent crimes, 18% for property crimes, and 18% for drug crimes. For jails, based on this source [1] I estimate 18% of the jail population is in for violent or weapons offenses; 3% for weapons; 30% for property crimes; and 31% for drug crimes. This source [2] suggests around 18% of federal prisoners are in for violent crimes; 17% for weapons/explosives/arson offenses; 11% for property crimes; 46% for drug crimes; and 8% for immigration reasons.
In my original comment, for state prisons, I counted violent crimes, property crimes, and 1/2 of other non-drug crimes to estimate that 76% of the prison population did something genuinely bad. For jails the number will be lower, both because there are fewer violent offenders and because the non-violent offenses tend to be less severe; if we count violent crimes, 1/2 of property crimes, and 1/2 of other non-drug crimes, we get 44%. For federal prisons, if we count violent crimes, property crimes, and weapons/explosives/arson offenses we get 46%. We can weight those three numbers by the sizes of the different groups to arrive at an overall figure of 61% for all incarcerated people in the US. (Of course, these numbers depend heavily on what you consider a "bad" crime.)
So of the entire incarcerated population of the US, about 1/2 to 2/3 are incarcerated for something genuinely bad. While you're correct to point out that this is lower than the 3/4 figure for state prisons, Broken_Hippo is still wrong to claim that "most" prisoners are in for things like non-violent drug offenses.
[0] https://www.bjs.gov/content/pub/pdf/cpus13.pdf
[1] https://upload.wikimedia.org/wikipedia/commons/d/d3/Felony_S...
[2] https://www.bop.gov/about/statistics/statistics_inmate_offen...