Unsurprisingly, several commenters point out that 'it's Down syndrome, not Down's syndrome'.
More interestingly, others express disappointment with the BBC for not using "person-first language" (also known, it seems, as 'people-first language'). One US-based commenter says:
These children are NOT "Down's syndrome babies" but instead babies with Down's syndrome.
Another, this time from Saddleworth here in the UK, writes:
People really should get used to the idea of naming people who have down syndrome as 'a person who has down syndrome' and NOT a 'down syndrome person' – they are people first and should not be called according to the condition they have, this is prejudice.
I'm not sure how I feel about this one. After, all, talking about 'British people' rather than 'people who are British' is not in itself prejudicial. I am almost certain the same is true for 'black people', 'young people' and so on.
And is person/people-first language more of an American phenomenon than a British one? I've never come across it before.