There are many different kinds of air purifier - I have some experience of the HEPA filter devices as in the hospital where I work, we've been using them throughout Covid in high risk areas as our Infection Prevention team are confident they remove at least some harmful materials from the atmosphere.
How much help they'd be in your particular case depends on what triggers OH's asthma, the air throughput of the device, how well maintained the filter itself is, and where the air input and outputs on the purifier are located.
For context, my contact with these devices during Covid has been based around a process I've been working on which puts a very fine, bitter tasting but harmless solution into the atmosphere - droplets the same size as Covid droplets. The filter units we used were effective at pulling these bitter droplets out of the air, but in a room probably 30 feet by twenty, it needed four units slightly bigger than a caravan fridge to keep the air clean.
In summary - based on my experience I'd expect a HEPA filter unit to be some help in removing some asthma triggers from the air - whether it would provide enough help to be worth having depends on the factors I described in paragraph 2.