The Kansas Dept of Corrections has their inmate classification manual (PDF only) available online.
Generally, there are certain automatic determinations (e.g., you have more than 50 years remaining to serve, you must be in max; you have an immigration detainer pending, you must be medium or higher). Otherwise, it’s on a points system. Your most serious current conviction is rated: 1st degree murder is worth 9 points, while shoplifting is worth only one. Then you get extra points for time left to serve (more time = more points), previous criminal history, a history of escapes or escape attempts, your age (younger = more points), etc. When they calculate the total, 17 points or more = max, 12-16 = medium-high, etc.
In Kansas, most inmates in protective custody are there because they asked to be there (they “checked in”). If there is an incident where one inmate attacks or threatens to attack another inmate and staff become aware of it, they’ll offer the victim the opportunity to go to PC, or ask them to sign a waiver if they don’t want to go.
Inmates who are particularly vulnerable or particularly problematic are sometimes sent to other states to do their time–most states have reciprocal agreements to house each other’s problem children.