In many young animals the intestine is set up to collect antibodies present in maternal milk and to pass it into the bloodstream to provide so-called "passive immunity" while the infant animal's own immune system becomes established. But, we don't believe human babies do this to a great extent*. This is not to say that the antibodies in breast milk are useless though. Far from it; the bulk of breast milk antibodies are what we call IgA; these are secreted antibodies and specialised for defending body surfaces. After the infant consumes them they adhere to the gut wall where they form a strong line of defence against a range of pathogens.

The major way that antibodies get into a young baby are via the mother's placenta, because once a baby goes beyond about 28 or 29 weeks of gestation, the placenta turns on special receptors called FC receptors which grab from the mother's bloodstream antibodies of a class called IgG, and they put them into the baby's bloodstream and that way, when the baby comes out, it's 'passively protected'. In other words, it's got antibodies recognising all the things that its mum is normally being exposed to in the environment in which she lives, so it's a good way of protecting the baby for its first weeks of life because those antibodies circulate for up to six months; after that time the baby should be able to stand on its own two immunological feet.

Unfortunately, although antibodies cannot cross the newborn gut wall, this is not true of everything, and some viruses, including HIV, can infect an infant via this route. In fact, the risk of a baby catching HIV from a mother if it's born vaginally and at term is actually quite low - about 5%; but if the mother breastfeeds the baby the risk can go up quite significantly - to over 20% - so not breastfeeding in those circumstances can actually spare quite a lot of infections.

[Note added later: I'm grateful to Michael Lindenmaier for writing to me about this question and also sending me this reference, which is pertinent to the subject].

"Differing from humans, IgG from breast milk in many animal species (rodents, bovines, cats, ferrets, etc.) are transported across the intestinal epithelium into the neonatal circulation. This transport is located at the duodenal and jejunal level where enterocytes express a surface membrane receptor able to bind Fc of IgG and to facilitate transcytosis of these immunoglobulins. Fcγ-R, which is very similar to the placenta receptor responsible for active transplacental transfer of IgG in humans, binds IgG but not other isotypes. Maternal milk antibodies represent an important part of circulating IgG in these animals, as they are involved in the negative feedback of endogenous IgG synthesis. This phenomenon stops abruptly as soon as weaning takes place. Neonatal calves that have a defect in such transfer of maternal immunoglobulins are at high risk of systemic infectious diseases.In humans, in whom gut closure occurs precociously, breast milk antibodies do not enter neonatal/infant circulation. A large part of immunoglobulins excreted in milk are IgA that protect mainly against enteric infections. The specificity of maternal milk IgA is driven by an entero-mammary cell circulation. Human milk also contains anti-idiotypic antibodies capable of enhancing infant antibody response. Maternal milk antibodies coat infant mucosal surfaces and some have a clear protective role."

From: Transfer of antibody via mother's milk, Philippe Van de Perre (2003), Vaccine 21: 3374-3376

*Note that, in an interview with Caltech scientist Pamela Bjorkman in 2008, we heard how some IgA is picked up and moved into circulation by intestinal cells.