He placed the baby into the crib, easing her down gently to avoid waking her.

easing 是动词 ease 的现在分词形式。ease 本意是"减轻、缓和"，在这里表示"缓慢而轻柔地移动"的动作。easing her down 指"轻柔缓慢地放下她"的动作。