As long as you keep them clean you should be fine. It's their poop which 'can' become infested with a bacteria. So just keep the bottom of the cage clean and all should be great.
I am 69 year old woman, average size 12/14, and was just diagnosed with Copd and have Aeortic regurgitation stenosis. My pulmonologist put me on Spiriva and oxygen. My question is that I have 3 cockatiel birds. Will I have to give them up? Of course my health comes first. They do create quite a mess as well as dust and feathers in the air.