I'm asthmatic, and can confirm that mould is not a good thing to have in a property for myself and fellow-sufferers.
I'm currently living in a ground floor apartment next to the Thames. There's some rising damp from the river; the unit above me had a flood in their kitchen several months back, and it's still drying out; and there was a damp patch on the other side of a wall to my place. The property's managing body said that there wasn't much that could be done about that since it was down to old pipes, but they can complain when I air washing outside.
The flat has a mould problem. There aren't any patches on the walls, but it'll start to smell fusty after a few days if it's not aired properly. Normally I keep the windows open if the weather's good, and run a dehumidifier if not.
However, the damp was sufficient to allow mildew to take root in a nearly new mattress. And, yes, I did turn and air it. Furthermore, several items of clothing went mouldy whilst I'd been away for a few weeks.
Whilst cleaning up after the latter I had an asthma attack. For a couple of days it was a struggle to breathe, and I popped out to the local clinic to get checked out.
For the next week I was completely wiped out. Ever noticed how old people tire easily? That's because their lungs don't work so well, and they can't get sufficient oxygen. Well, the aftermath of the attack left me in that sort of state, and I didn't feel up to much beyond walking to the shop at the end of the road.
Whilst in some cases mould is the fault of a tenant, in others it could be due to damp proofing problems in the building. In which case it'd be the owner's responsibility to fix it.