There are several times when I've seen young children apparently on their own, and in each case I try to check around to see where their carer is. If this takes a while then I'll observe from a distance as long as it takes to be satisfied that the kids are in no danger. It's quite right and proper for people to be concerned about possible risks to children.
On the other hand, my observations instantly end when (as always happens) the kids' obvious carers show up. It doesn't take long to tell who they are because the children are comfortable around them and there's nothing forced about the interaction.
Common sense tells you that the same thing should have happened in this case: regardless of what prompted the concern (and, let's face it, Naomi's blatantly admitted that the overriding key factor for her would be the mismatch in skin colour, which is by definition a racial judgement), then it should take only minimal observation for that concern to be addressed from a distance without ever having to interfere.
As to being polite and saying "thank you", while perhaps that was preferable, it's ludicrous to suggest that the onus is on the babysitter to have to explain himself. It might be noted that racial and sexual profiling is rather more common than some people care to admit -- I shouldn't wonder if after the one hundredth time of having to say "thanks for letting your racist concerns prompt you to take unnecessary action, as would have been clear if you'd actually been paying attention", it would have been harder to maintain the facade of being grateful that people see your skin colour and instantly decide on that alone that there's something fishy about you.
There are so many things the woman could have done instead to avoid the situation; since, by definition, it started when she decided to intervene, it is first and foremost up to her to shape the interaction. That could be either by watching the children discretely and realising that they are happy, which should lead her to the obvious conclusion that this man was known to the kids and they were happy around him, and therefore no further action need be taken.
Easy -- if you think about it. And, if you don't, and just act on prejudices, then it's the very definition of racism. Dress it up however you like; if you base your actions primarily on somebody's race then you are a racist.