This isn't true. Back in the 80's and prior, the US Gov't gave even illegal immigrants Social Security numbers. In fact, they issued driver's licensces and all. Also, when one becomes naturalized, you pay a processing fee not a fine.
Ginatio, I can speak from first-hand experience that there is no fine, just a fee. I was born in 83, came into the country in 86 and was issued a SSN.