No (or a bit more scientifically: very very unlikely)
There a couple of reasons why your idea would not work:
Viruses are not alive, so you can't really kill them. The only ways to get rid of all airborne viruses at once would be to either:
- A) UV-sterilize the whole atmosphere (bad idea, good xkcd what-if question though) or
- B) fill the whole atmosphere with something that replicates itself and consumes very small particles of biological material (oops, we released grey goo).
It's also not possible to create bacteria or viruses that can both spread quickly (which means it need to be able to grow / self-replicate) and is physically unable to mutate.
Mutations arise from errors in DNA replication, and DNA replication is never 100% accurate, you can reduce the error rate, but it will never reach 0, so mutations will always be possible. It's also impossible to predict the 'direction' of mutations so you can't exclude that they will be dangerous for humans either.
As other comments have already mentioned there are even more problems with the idea itself:
- not all viruses are airborne, many reside in animal hosts, so getting rid of the airborne ones will not do much
- many viruses are not pathogenic for humans but ecologically important, you will get an unknown amount of ecological impact