Is it dangerous for a child to breastfeed from a drunk mother?
It doesn't seem to be culturally acceptable, but if breast milk contains roughly the same percentage of alcohol as that person's blood, i.e. a few per mille at most, it doesn't seem like it should be dangerous. Assuming the mother doesn't drop the child.
Examples of this claim:
All of this answer is based on what I found on a recent review on the subject of alcohol and breastfeeding Haastrup, M. B., Pottegård, A. & Damkier, P. Alcohol and Breastfeeding. Basic Clin. Pharmacol. Toxicol. 114, 168–173 (2014).
First of all, it is interesting to address a common myth that alcohol ingestion can increase milk production and secretion. (The OP mentions that it is not socially acceptable to breastfeed while drinking, but it seems it is actually encouraged in some places.) There is no evidence to support this, and some evidence against:
One study of 22 lactating women, who were asked to express milk on two different days, one of which they had ingested 0.3 g alcohol per kg (corresponding to about 1.8 units of alcohol, assuming a body-weight of 70 kg and an alcohol content of 12 g per unit), showed that the amount of milk expressed was 9.3% lower on average in the first two hours after alcohol consumption (...)It has also been demonstrated that alcohol dose-dependently inhibits oxytocin and thereby the milk ejection reflex, which could probably explain this reduction in milk yield.
Of course, that minor reduction might not be relevant to the baby’s health.
Concerning the concentration of alcohol in milk, it is about 5-6% of the weight-adjusted maternal dose, so it is possible to calculate how much alcohol a child would ingest from a mother. In the review’s conclusions, we can read this, and I think this is closest to an answer to the question:
Assuming the worst possible scenario where a mother engages in binge drinking and ingests four drinks of 12 g pure alcohol and then breastfeeds her child at the time of the maximum blood alcohol concentration, the child would still not have a blood alcohol level of more than 0.005%. It appears biologically implausible that occasional exposure to such amounts should be related to clinically meaningful effects to the nursing children. The effect of occasional alcohol consumption on milk production is small, temporary and unlikely to be of clinical relevance. Generally, there is little clinical evidence to suggest that breastfed children are adversely affected in spite of the fact that almost half of all lactating women in Western countries ingest alcohol occasionally.
Finally, many studies are cited in this review about possible effects of alcohol breastfed to infants, but they are not conclusive. Some point to sleep disturbances, some say sleep patterns are normal. Some point to reduced milk intake, others say it is normal. There is even one who suggests delayed psychomotor development, but a replication showed no such effect. Most of those studies lack adequate control of confounders (like psychosocial adversities, for instance).
In conclusion: it does not seem that drinking while breastfeeding is directly harmful to the child.
