The problem doesn't come when the mother is a different ABO blood type from her babies, thankfully! The problem only comes when the mother is a different Rh (or sometimes other) blood type from her babies.
Say a mom is O-. If her partner is A+, and the baby is A+, she's probably going to gestate that baby without problems, and then deliver it without problems. The ABO antigen is too large to pass through the placenta, and even though the Rh factor antigen (the + and - in the blood type) is small enough to pass through the placenta, if she hasn't been exposed to an antigen previously, she won't have a negative reaction to it. So she'll deliver the first baby without problems.
After that, she'll probably gestate the next baby without too many problems, but upon delivery, assuming some blood transfer, there could well be immune reactions on both ends of the equation - there's the possibility of both the mother and the child (though the mother is at a higher risk) having a negative reaction to birth. She already has antibodies ready to attack the Rh "antigen". This is why Rho(D) immune globulin is administered when the mother's Rh factor is different from her baby's.
Until about 1955, the majority of [second or beyond] infants with Rh factors different from their mothers had either chronic health problems, or passed away within the first few days of life due to the anemia caused by the destruction of erythrocysts (erythroblastis fetalis) but since the discoveries of the 50s, the majority of people born or giving birth to those with different blood types, have not died from that cause.