I can’t say with any real certainty how safe it is, but my 12 and 10 year old kids have neopets. When they signed up they had to fill in their ages, they are not allowed to have neofriends or use the message board there.
Here’s what it says on the neopets website about safety.
The biggest problem I had with Neopets was the spam. In order to get certain rewards and things, you had to sign up for mailing lists. This was a couple of years ago, and I don’t know if they still do it, but I strongly disapproved of it. This may not be a problem for you if you kids have their own email accounts… but if they are using your address, then you will suddenly find yourself receiving marketing emails on all kinds of things that you probably have no interest in.
Neopets has a very strict filter on all neomails moving through their system. It filters out any salacious word you can imagine. One time I was talking to a friend about rubber nipples on baby bottles and the server wouldn’t send that. They also filter the message board pretty strictly. I wouldn’t be worried if I were you – at all. Truly. I don’t even think they can give out their email address through the neomail system. It’s very secure.
If you are worried, you can always email an administrator of the board and ask about their security precautions. But really I wouldn’t waste time thinking about it. It’s harmless.