Ok, I read stories about women from Mexico coming up into the US when the baby is due and giving birth, since that makes the child a US citizen.

So what happens now, the mother is not a citizen and just on a tourist visit.
Since her child is her dependent, can she get to stay in the US once the child is born? How would they deport her if she has a child who is a US citizen?

Having a US born child does not give the mother/parents legal status. There are 2 choices: move back to their home country with the baby OR stay as illegal aliens in US. The child will be eligible to apply green card for mother/parents when he/she is 21 years of age, employed with sufficient income (125% of poverty guidelines), must be able to submit tax returns of three years.

In case the illegal parent(s) get deported, they have 2 choices: take the child with them OR leave the child in US. Social Services will find them foster homes. If they choose to leave the baby in US, when the child is 21, he/she cannot petition green card for birth parents.

Then why are so many doing it ( at least according to the story I read) they said there was an entire industry bring pregnant women into US to give birth.

Is it just to make baby a US citizen ?