This option still exists in many places. Japan recently opened a modern version of the same thing. Calling it a baby drop box. Unfortunatly, the first "baby" was a three year old kid.
http://www.msnbc.msn.com/id/18709944/

Regardless of the above, yes the father has rights to the child, but for whatever reason his rights are often not followed up on.

If this gives a mother and child an option for a better life, I say give them the chance.