I think the word "safe" is being used here in a context I am not familiar with. If he's referring to safe in all humans, and no human of any age in any condition would never have an adverse reaction at all, then I suppose I agree.
Stillwagon points out (perhaps by splitting hairs) that no vaccine prevents infection, since even the vaccine prompts an immune response, and then of course the body is supposed to produce and store memories of antibodies to the vaccine. He also discusses adjuvants, preservatives and inflammation response as he describes how vaccines work. This has nothing to do with "safety" per se.
I immediately thought about smallpox. This one has a high kill rate. I'd much prefer to have a primed immune system in case I caught that nasty virus.
Perhaps the other chemicals in a vaccine vial need some attention.