I'd always sort of thought that people with severe peanut allergies could have a reaction from being too close to peanuts, even without touching them. Turns out, that's not true. Usually.
As Canadian allergy doctor Antony Ham Pong explains, most of the rules preventing bringing peanut products into a school or house where peanut-sensitive kids spend time aren't about sharing airspace with peanuts. Instead, they're meant to prevent kids from surreptitiously sharing peanut-laced food products and to prevent younger kids from smearing peanut butter on toys that could end up in the mouths of kids with peanut allergies.
That said, though, Pong points out that there are situations where a peanut allergy can be triggered by the air you breathe.
If a peanut allergic person breathes enough of the peanut protein in the air, the person can have a serious allergic reaction, asthma attack or anaphylaxis. Situations in which this can occur are unusual but can happen. For instance, if a large number of people are opening packages of peanut at the same time – e.g. when peanut packages served on an airplane – and the peanut protein dust gets into the air in an enclosed space.
Other examples would be boiling or frying a food with peanut, as minute peanut particles can then get into the air [through steam or oil particles carrying peanut protein]. Another example could be a floor with large amounts of peanut shells and containing peanut dust where people walking on the shells can stir up peanut dust in the air. (An example would be sports bars.)