I don't know if it's a must to have a SIN card for your child but I think since the SIN card is one of the very important identity here in Canada so you might as well apply it. Cuz you'll never know when you'll need it in the future.
SIN is only for the tax purpose. My older one is 14 now, and he is in Canada, he does not have SIN, but I am going to get him one soon, I want him to do some part time job.
If the SIN is free and can apply by mail, why not?
As Yukimommy has said, she needs it to apply for RESP, so it is always good to have all documents ready.
As far as I know, the parent has to come with legal documents in face in Canada to apply SIN card. The department is 'Labor department' so it cannot apply through Consulate.
The SIN will card by registered mail after one month.