Maybe someone with more economic insight can clarify the issue, but my understanding of subsidies is that they're ultimately harmful to the economy of the country providing them. Obviously the recipient industry and its owners will benefit immediately from subsidies they receive, but the economy as a whole -- and the taxpayers who pay the subsidies -- will be harmed in the aggregate more than anyone will be helped. Ultimately, economies based on subsidies (i.e., Communism) cannot compete with free markets, but if subsidies are kept small as a proportion of the economy as a whole then specific industries can be built up without seriously threatening the survival of the rest of the system.
So, European Airbus subsidies hurt Europe more than they hurt America -- in total -- but they can do significant damage to the American commercial airline industry, and Boeing specifically. America doesn't want to hurt itself by subsidizing Boeing, but we also don't want Boeing to be put out of business by an opponent that doesn't mind hurting itself to hurt us. Aside from larger economic concerns, we have a national security interest in maintaining a domestic commercial airplane manufacturer, so we can't just sit idle while Boeing is put out of business.
There doesn't really appear to be any way to fight subsidies without using subsidies. The only comfort is that whichever side uses fewer will end up stronger economically in the long run.