Ben kidnapped Alex because he refused to follow orders to kill an innocent child and instead chose to raise her as his own daughter, also sparing Danielle.
Defiance of Widmore's Orders
Ben's decision to kidnap Alex was a direct act of defiance against specific instructions he had received. Widmore, a prominent figure, had commanded Ben to kill Alex, who was then an infant. However, Ben found himself unwilling to carry out such a heinous act against an innocent child.
A Choice to Nurture
Instead of complying with the order to eliminate the child, Ben made a pivotal choice. He neglected Widmore's orders and abducted Alex with the intention of raising her himself. This action served a dual purpose: it saved Alex's life and also allowed her mother, Danielle, to live. By taking Alex into his care, Ben effectively adopted her, nurturing her as his own daughter and shaping her upbringing outside the destructive path set out by Widmore.