You could prove they don't exist by how they are used and developed in history, their usage by the people who believe them, observation, and pretty much comparison to that and things that are psychological and physiological in nature. You can also cross-reference various religious beliefs from different gods and notice a pattern of human need to see the unknown something they can describe whether by practice, faith, involvement, so have you.
Not something you can test in a test tube. Just "god does/doesn't exist" rather than I think, I believe, I have faith, would have different criteria for whether it's fact or not. When we find historical artifacts from my knowledge (unless maybe it's the blood crying statue) we don't use science to determine if the events were true or not. Yet, we find conclusions based on strong evidence and historical sequence to fill in the blanks that we conclude it is true. Even when we did not exist back when to confirm its 100% true.
I'd say fairies go by the same logic. But then we don't tell our children they "could" exist, I'm sure. Even Santa. When I grew up my parents told us later down the line that they were Santa (I don't think we believed they were real that long. They had bills to pay). It didn't ruin the intent and fun, though. The purpose (like god) wasn't to think about its existence. So, that's the only comparison I see.