Which is why any physical therapist would recommend a person sit down when using these bands if the person is unstable on their feet. I have used these bands and have only had problems when using them incorrectly.
The article said he was doing some sort of rowing exercise, which would imply he was sitting in some fashion. The article doesn't say, but I suspect he was sitting on a chair or stool. It would be hard to fall when your rear end is on the floor.