A sensible person is one who exercises good judgment.
Literal
Discernment-having person [quotative-という] [possessive-の] [topic-は] good-sense [object-を] employ person is.
~というのは…である is a definitional pattern: 'what we call X is Y.' 良識を働かす means 'to exercise common sense' — 働かす is the causative of 働く (to work), here meaning 'to put to work/employ.'