If a parent is an alcoholic, there is a strong probability that the child will also become an alcoholic.
Statistically, the child of an alcoholic is several times more likely to be alcoholic him- or herself than children not growing up with alcoholic parents. Scientific studies have pointed in the direction of a genetic link between parent and child that makes the latter a strong candidate for addiction to alcohol. That does not mean that the child who shares the specific gene with the parent that influences addictive behavior is condemned to be an alcoholic; it does mean that, absent prevent behavioral measures, the child will likely grow up to share the parent's addiction.
In addition to the genetic link involving addictive behavior and alcoholism there are also environmental factors that influence whether a child will follow the parent's path with regard to alcohol abuse. A child who grows up with at least one alcoholic parent, especially when that parent is a functioning alcoholic, is more likely to become an alcoholic him- or herself due to the protracted and highly influential exposure to alcohol consumption in which the child is immersed while growing up. Environmental influences on a child's mental development are enormously important, and an unhealthy environment involving alcholic parents is extremely likely to affect the child's behavior.
Because of the prevalence of the genetic link between the parent and the child, and because of the importance of environmental factors in a child's emotional development, the children of alcoholics are far more likely to become alcoholics themselves.