The particular episode that used Sinnerman, "Sins of the Father" (1998) had a storyline about a family with a history relating to slavery and the Confederacy.
Another website says the song was used in an episode of "Miami Vice", though don't recall it myself.
The song is an Afro-American spiritual, used at religious meetings to evoke the day of judgement and the idea of the "sinnerman" trying to find a place to hide from the end of the world and God's judgement. The lyrics reference biblical descriptions of the end of the world - rivers running with blood, sea boiling etc. Nina Simone was supposed to have heard it used this way by her mother, who was a Methodist preacher.
In the various cop show / movie scenarios, the song has typically been used to evoke /accompany a hunted bad guy trying to escape from retribution/justice (one of the classic crime show/movie scenarios), or (more rarely) to atone.