Cameras have it built into them. It's called image stabilization. Beyond that you can do some correction in post with digital stabilization. aka deshaker. And various other gadgets to help the camcorder out at the point of capture. Heavy tripod, fancy heads for super smooth pans and such. Monopod, shoulder rig, remote only operation and other things to take the shake makers out of the picture.
It's taking a while to download over dialup. But your codec is already suffering from stabilization issues in the first 7 seconds. You could get a wide angle lens and get closer to the subject to take some of that issue out of the picture. Or just don't use zoom, or as much zoom. It looks like the heart beat is translating through the tripod, or the platform/stands are being shaken by the people on it, or the PA system. If it's the heart beat, don't touch the tripod if you're not adjusting anything. If it's the stands going with a shoulder mount rig might allow for the cameramans body to act as a stabilizer.
I don't know of any shockmounts for cameras like you can get for microphones. Some cameras have decent optical image stabilization in the lens, but it can only do so much. Some cameras use a combination of optical and digital on camera to improve things. Normally the better the camera, the better the end result. But not all things are created equal.