The phrase "balanced approach" refers to a method or attitude that considers and weighs multiple factors, opinions, or viewpoints before making a decision or taking action. It suggests an even-handed and rational approach to problem solving, where all relevant information is taken into account in order to arrive at the best possible outcome.
When referring to "one's balanced approach," it means that this method of thinking and acting applies specifically to that individual. In other words, they are taking a considered and well-rounded perspective when making decisions or tackling challenges. This phrase can be used in a positive light to describe someone who is thoughtful and measured in their actions, or negatively to suggest that someone is overly cautious or indecisive.