Hmm babies eh? Totally normal, I'm not aware of any carrier who bans revenue earning humans of any description. I think you will find they also allow loud businessmen, drunks, and people getting up and down to go to the loos all the time too!
A lot of ex-pat management in the Middle East have family travel perks written in to their contracts so yes, you see plenty of kids in there. My record was 4 young kids travelling together in J on Etihad. Looking thoroughly bored.