i have found that most people, when you tell them "here record this pass" or whatever. dont really care about the quality of the video, so they just point the camera in the general direction and hit the record button.
some of the problems i have seen with shakey videos are, holding the camera with one hand. some people can do it and make the video nice. most cant. whenever i shoot videos i hold the camera with one hand, and steady my arm/camera hand with my other hand.
people zoom in too far, and leave the camera zoomed in. when the camera is zoomed in every movement is picked up 10x as much. a little shake can ruin the entire video. plus people over use zoom as it is. we dont need to read the tire size as the car is staging.
if the camera has an lcd screen (most if not all do these days) i have found the best way to record a good video is to USE it, if im dedicated to recording something, then my interest is in making the video look the best i can. i will watch the race through the lcd as im recording it to make sure i have good angles, zoom, etc. ALOT of people that take crappy videos, watch what they are recording while not thinking about what their hands are doing, they may have the camera pointed towards the race, but they are watching it with their eyes and not thinking about what the camera is seeing, then the video ends up turning out like shit because its all shakey.
these are a few things that i have seen, and that i pay attention to while im recording something. hope it helps