Why would one consider feeding a child anything other than breast milk in the first place? If it were my kid, it wouldn't matter if it's from a goat, cow, or whale for that matter. I'd be keeping him or her away from all of them.
Assuming the mother is unable to breastfeed, the replacement needs to be chosen carefully. (In some communities, there are breast milk banks, an option if a mother's own production is insufficient.) Goat's milk is not without risk, even if it's the basis of a prepared infant formula. (The same applies to cow's milk.)
The argument that one is more allergenic than the other doesn't wash. The American Academy of Pediatrics also recommends against feeding small children cow's milk, because it increases the risk of iron deficiency anemia, which at that age that can have lifelong consequences. For once, the medical establishment seems to have got something right.
I'll cite a 2010 paper in the journal Pediatrics describing severe metabolic disturbances in a five-month old. The infant was receiving raw goat's milk; pasteurization takes away the infectious disease risk, but does nothing for the nutritional composition:
Fresh Goat's Milk for Infants: Myths and Realities—A Review
The infant in this report presented with severe hypernatremia and azotemia in addition to other electrolyte abnormalities. Goat's milk contains 50 mg of sodium and 3.56 g of protein per 100 mL, approximately 3 times that in human milk (17 mg and 1.03 g per 100 mL, respectively).6 The estimated requirements of sodium and protein for infants <6 months old are 100 to 200 mg/day and 9 to 11 g/day, respectively.7 The infant described here was receiving ∼500 mg/day of sodium and 30 g/day of protein, with a total intake of 32 oz of goat's milk per day. The immature kidneys in very young infants have difficulty handling the byproducts of foods with a high renal solute load.8 Sodium excretion capacity matures more slowly than glomerular filtration rate and does not attain full capacity until the second year of life.9 Therefore, infants fed fresh goat's milk are at substantive risk for hypernatremia and azotemia, particularly in the face of dehydration (as in the case described here), which may in turn result in major central nervous system pathology, including diffuse encephalopathy, intraparenchymal hemorrhage, or thromboses10 as manifested in our patient...
...The main benefit claimed by proponents of fresh goat's milk for infants is that it is less allergenic than cow's milk and is a suitable substitute for infants who are allergic to the latter. However, evidence shows that most infants who are allergic to cow's milk are also allergic to goat's milk. In vitro studies have shown that there is an extensive cross-reactivity of sera from individuals who are allergic to cow's milk with proteins found in goat's milk.17,–,19 In 1 study, 26 children with immunoglobulin E–mediated cow's milk allergy also had positive skin test responses to goat's milk, and 24 of 26 had positive double-blind, placebo-controlled, oral food challenges with fresh goat's milk.20 There have been case reports of severe life-threatening anaphylactic reactions after the ingestion of commercial goat's milk preparation in infants with documented cow's milk protein allergy.21 Furthermore, infants and young children may have signs, symptoms, and serology positive for goat's milk without being allergic to cow's milk.22,–,25 In a retrospective study, children presented with severe allergic reactions, including anaphylaxis, after consumption of goat's milk products but tolerated cow's milk products.26
Folate deficiency with anemia in
infants fed homemade formula based on
goat's milk has been described.27,28
In fact, “goat's milk anemia” was the
name given to the macrocytic
hyperchromic megaloblastic anemia
observed in infants fed goat's milk in
Europe during the 1920s and 1930s.29
The anemia was thought to be more
severe than that associated with
exclusive cow's milk feeding and was
cured by giving supplements of liver
extracts. The concentration of folate
in goat's milk is 6 μg/L in comparison
to human breast milk, which contains
50 μg/L.30 Infants younger than 6
months of age need 65 μg/day of
folate, and the recommended daily
allowance increases with age.30
There have been reports of infections
such as Q fever, toxoplasmosis, and
brucellosis associated with feeding
raw goat's milk.31,–,33 Consumption of
unpasteurized goat's milk has also
been implicated in the development of
Escherichia coli O157:H7–associated
hemolytic uremic syndrome.34,35
Although raw goat's milk is a proven
vehicle for pathogen transmission, the
belief persists that raw dairy
products are healthier and that
pasteurized products are less
beneficial and even harmful.5
The best way to go, in my humble opinion: maximize Mom's health! Make sure she's vitamin D replete, is getting plenty of calcium, magnesium and iron (lactation nutritionally hammers the mother), is eating well with lots of variety, and is in a stress-reduced environment (the regular stresses of parenting notwithstanding ;) ). This increases the chances that milk output will be sufficient and of sufficient quality to keep the baby happy and healthy.