How do you prove you don’t pick your nose?

You can’t.

You can not prove something that never happens, does not happen. It’s called a universal negative and can’t really be proven. It’s like proving god does not exist, you can’t prove she does not exist. You can simply notice the « absence of evidence » she exists.

So if someone says about you « This guy picks his nose » the only way you can prove you’re not is to continue living your life without picking your nose. You smile at the people observing you in the hope to finally see you pick your nose and ignore the uncomfortable thought of being watched by those people.
Even if those people tell other people to try to catch you pick your nose, even if that goes on for years and years, even if you end up thinking you're crazy, taking medications and spending time in the hospital because of that, all you can do is live your life (in a slightly uncomfortable manner) until it becomes globally accepted that you, indeed, don’t pick your nose.