For example, more people are choosing to live together and have children without getting married.
Literal
For-example, marry-not [nominalizer-で] cohabit children [object-を] have people [subject-が] is-increasing.
~ないで ('without doing X') uses the negative + で connective — 'without marrying.' もうける (儲ける) has a second meaning 'to have (children),' distinct from its 'profit' sense. Social-change/commentary register.