Using the “Single Responsibility Principle” forces my containers to have public setters

I'm trying hard to design following the SOLID principles. What I've found is that when you use the "Single Responsibility Principle" (the S of SOLID) you usually have to split classes between the data containers and the data processors. For example If I have a class person with 5 properties that is read from DB instead of putting everything inside a class I create a Person class with the properties and another PersonReader class that reads that information from the database and creates the Person.

If I do that I have to open the Person properties so PersonReader could access them but then I have less encapsulation than putting everything inside a black box and making the properties only readable.

Am I missing something or is this a drawback of this principle?

Thanks in advance

EDIT: I've changed the person writer to a person reader because there was no need to make property setters public at the beginning.