Hand held camera animation can be difficult to reproduce in 3D. Randomly animating keyframes never seems random enough, and will eventually paint you in a corner when there is a need for changes.
There are a few easy ways that will give you a natural and realistic look. Here, Gleb Alexandrov demonstrates seven easy steps for creating that coveted look by creating an easy setup for handheld camera animation.
A good rule of thumb is that anytime you want to move something in animation, a concise and practical rig should be the thing that holds the keys.
In-line with that way of thinking, Gleb shows how setting up a simple parent chain can give you some really nice control over your camera in Blender.
The handheld camera animation comes from using procedural noise to drive the motion.