The maintenance crew is immune to attacks, and can run faster than the player. They have access to every door, even the medic & guard-only doors. Like the medics, they will not take notice of anything happening around them. For example, robbery from maintenance desks, excavation in maintenance rooms, slumbers in maintenance beds - the maintenance crew will ignore all this behavior.
Specific behavior depends on what role they fill.
If a guard is able to report prison damage to the control room, a repairman will be sent to repair that damage (plodding at walking speed). This could be anything - i.e., broken walls, tunnels, unscrewed vents, flooding toilets, etc. Once they are done with the repair, repairmen will run (full-speed) back to their room. Repairmen will normally stay in their own maintenance room, comprising of a maintenance desk, a toilet, and a bed. Maintenance rooms are usually guarded with utility doors.
Note that when the Damage Repaired message pops up, it marks the time when a repairman is called upon, not when the damage is actually fixed. If "Damage Found" is displayed, it means that a guard has seen actual damage to the prison and will go report it. If "Damage Repaired" displays with no preceding message, it signals a flooded toilet being repaired. However, as maintenance staff are invincible, once the Damage Repaired message shows, repairs cannot be prevented.
Apart from repairmen, there are also other types of maintenance people. The warden and interviewers may not be maintenance people in the actual prison, but the game considers them such as they are marked as maintenance staff on the map.
There are the interviewers, whom you can meet at the job office at job time for $10 (this will only work if you are unemployed).
The warden runs the whole prison, attends Roll Call in some prisons, and can be seen in their office or wandering around.
A few maintenance workers regulate matters at the visitor's room. Librarians organize the library books. Some maintenance employees will hang out at the social room to cheer you up from prison gloom. These types of NPCs have generic names (Bob, Kathy, etc.).
Chefs in the kitchen will cook the meals. Do note that they have no effect on the Kitchen Job. Their names are preceded by the word "Chef".
A repairman will typically wear a hard hat along with grey overalls and a red shirt under it. Sometimes they will wield a plunger when they are fixing a toilet. (They will not hold a hammer even if the repairman on the Damage Repaired message has a hammer). Other maintenance members will wear a grey shirt (with some color variation depending on their role). Chefs will obviously wear chef outfits and wardens have custom outfits (and names) themed for each prison.