My understanding of the breeding of horses (and someone correct me if I'm wrong) is that while "the act" may be supervised by humans it must happen the good old fashioned way in order for the colt to be eligilbe to race.
There shouldn't be any Barbaro offspring since he won't be around to sire them.
You are right. Jockey Club does not allow artificial insemination.
As far as litle Barbaros, I believe the Jacksons have a full brother to Barbaro, and his broodmare, La Ville Rouge, is in foal to Dynaformer again.
TMI: It is true that for Thoroughbred horses to race, they must be the result of live cover (i.e., no artificial insemination allowed). However, A Thoroughbred stallion may be collected and used to artificially inseminate Quarter Horse or Paint or Appaloussa mares, and then the offspring may be raced in their individual breed races. But a horse must be a registered Thoroughbred (and thus conceived via live cover) to race in Thoroughbred races.
As stated before, though, given Barbaro's condition, it would not have been possible to collect his semen.