To be able to access the Prison Depths, you'll need to have gained the Spider Rune.
In Promenade of the Condemned, towards the end of the map, near the third/fourth lift it'll take you to a place which can only be scaled via using the spider rune.
Prisoners Quarters -> Promenade of the Condemned -> Prison Depths.
I've linked a video below as well as a full playlist showing you how to get all areas/runes. (If you need to get Spider Rune as they are all in order)