This is a rare, congenital disorder of baby goats known as "palatoschisis"; but if you raise enough goats it is one that may occur in your herd. For this reason, it is a good idea to check every baby for this problem at birth. Upon careful examination of the roof of the mouth (forget about the technical terminology), you will notice a thin slit toward the back. Another thing to watch for is that this anomaly is sometimes accompanied by other congenital malformations (cleft lip, rigid limbs).
The first thing you will notice is milk dribbling from the nostrils after nursing. Occasionally, there will be some choking or snorting when they nurse. Later, respiratory infections may develop due to inhalation of milk or food.
This is one of those situations where you need to make some very quick judgments. Surgical repair can be performed at about 3 months by a skilled veterinarian, but this will be quite expensive. If you have an outlet for selling kids for meat consumption, then it may be possible to raise the kid for a few months. Definitely, NEVER keep the animal for breeding. If you want to try to raise it, remove it from the mother and raise it on a bottle. This, of course, means that you will need to bottle feed colostrum right away. You can make a "prosthesis" out of thin rubber (like a bicycle tube) to attach to the bottle that will cover the slit in the roof of the mouth. It will take a little experimentation to make this all work, but it is possible. You probably should try to wean the kid at between one and two months of age. Some of these animals can handle hay and grain with little difficulty; others will snort and cough and get the food up into the air passages; these probably should be destroyed. We have raised hundreds of baby goats and have only had one instance of this ailment. Fortunately, we were able to raise it for meat purposes by the method above. It is up to you to make the initial hard decision as to whether or not to try; just be sure to make the decision early to avoid a lot of hardship on you and the baby. Some say that the condition is more likely with inbreeding. Also, I have heard that some sources say that selenium deficiency in the mother increases the risk; I don't know if this is true or not.