Which sensor is better for video, cropped or full frame?
It depends.

All of the questions at http://photo.stackexchange.com regarding the relationship between sensor size and Depth of Field (DoF) apply to video as well. (See, for example: https://photo.stackexchange.com/q/12098/15871, https://photo.stackexchange.com/questions/7904/how-does-background-blur-bokeh-relate-to-sensor-size, and especially https://photo.stackexchange.com/questions/3986/when-do-the-differences-between-aps-c-and-full-frame-sensors-matter-and-why)   If you want very shallow depth of field a FF will give you shallower DoF for the same subject distance, field of view (FoV), and aperture than a  cropped sensor will. This is because the FF camera must use a longer focal length than the crop sensor to achieve the same FoV from the same subject distance. If you want very large depth of field the cropped sensor will have the advantage over the FF camera with the same distance/FoV/aperture.

I think what the comment you heard might have been addressing was the difference in resolution that once existed between most cropped sensor cameras and most full frame cameras. Just a few years ago most APS-C cameras ranged from about 10-12MP while FF cameras where at around 18-20MP. This is not so much the case any more as improvements in sensor technology have allowed APS-C cameras in the 18-20MP range. Since 1920X1080 HD video is only about 2MP per frame, any camera in the 10MP+ range is going to downsize the output of the sensor to 1920X1080 so there is not much difference between a 12MP APS-C camera and a 20MP full frame camera in that regard.