Football players are taught in the open field to hold the ball in their dominant arm. The more coordinated and stronger arm is at less risk of allowing a fumble. In running down the sideline or anticipating contact, the ball should be in the arm closest to the sideline and away from the defense so that a fumbled ball likely will roll out of bounds. In summary, the ball alternates between arms for sole purpose of providing the best ball-protection for the situation at hand.
My wife, in holding all of our four children over the years, always has preferred carrying the youngsters in her left arm. She, like most of people, is right handed and therefore argues that carrying in her left arm frees up her dominant right arm to otherwise help the baby or perform other tasks. She argues that the baby is in the arm that provides the best security and functionality. This would suggest that carrying a baby is completely determined by handedness. If the mother wants her dominant hand free (like my wife), right-handed mothers would most often prefer holding the baby in their left side, and left-handed mothers would prefer holding in their right side. If the mother feels more secure holding with the dominant arm, right-handed mothers would prefer the right side, and left-handed mothers would prefer their left side. Simply put, carrying a baby is just like carrying a football.
The literature suggests, however, that 70-85% of mothers, regardless of handedness, prefer holding their baby on their left side. This “cradling bias” is present across numerous mammalian species, which indicates an evolutionary origin. The right side of our brain does a better job in spatial coordination and social learning. Since the right brain visually processes stimuli in our left visual field, having the baby in the left arm allows both the mother and the baby to have each other in their respective left visual fields when face-to-face. This allows both to maximize processing and spatial position/interplay of each other by using their right hemispheres.