"No, they do not have babies."
Translation:Nein, sie haben keine Babys.
Yes, I guess the problem arises partly because of marketing. Marketing people tend to use a lot of English phrases and prominently place their slogans everywhere. So, it's not uncommon to see 'Babies' written on a pack of disposable diapers. Many Germans adopt these spellings. But as christian already pointed out the correct German plural is 'Babys'.
I guess there is a subtle difference here: "Sie haben keine Babys" negates "Babys", and "Babys haben sie nicht" negates "haben". As if they had something, but not babies.
Normally, if you use the straight word order and say that someone does not have something (or has "no something"), you use "kein". In English, "no" is not so commonly used as "kein" in German.
Also, I believe it is a thing you learn by seeing it numerous times. After you get tons of "nicht's" and "kein's", you get a feel of which word suits better.
I'm not exactly an expert on this, but I'd say it is so uncommon that it is almost incorrect. Basically, "nicht" negates a verb, and "kein" negates a noun, but in this case you can negate both the verb (have) and the noun (babies). I can suggest a rule of thumb: if you can fit in "kein", use it (and negate the noun). If you can't, it's time to use "nicht".
@Pronkman666 means the German plural. In a muliple choice question, Duolingo asks us to translate "No, they do not have babies" to German, but gives us "Nein, sie haben keine Babies" as the only correct option (as at 06.05.2020). Picture here: https://drive.google.com/open?id=1yASj0l4wS-fhGYbTHYJ2LEApFeZCMKPf
@Pronkman666 means the German plural.
The lowercase probably threw me off -- the German word is, of course, Babys with a capital B.
gives us "Nein, sie haben keine Babies" as the only correct option (as at 06.05.2020). Picture here
Wow, that is bizarre. I'm not sure where that came from -- I can't see anything like that on the back end.
Lingot for the screen shot, though.