This is certainly a very topical issue on the minds of a lot of people these days, though not the minds that many would probably think at first: to the average traveler, the plane's maintenance history is (sadly) an afterthought. While travelers will all of course say that, unquestionably, their paramount #1 priority is safety above all else, those abstract notions break down when you look at how customers truly think about air travel: for the vast majority of passengers, the most important factor is price, and the only time when safety even factors into their decision-making, in my experience, is in the event of an extreme or severe circumstance (i.e., when an airline is particularly known for bad maintenance or a poor safety record).
Since air travel in the U.S. is remarkably safe (statistically the safest way to travel, by far), most customers don't even think about safety, as it is just assumed (rightly or wrongly). The way that we can conclusively prove that the issue of civil aviation maintenance outsourcing in the U.S. is a hot topic with politicians and unions, but not with customers, is the behavior of consumers in response to continued public and media attention on the issue: with all the articles, reports and news stories about maintenance outsourcing, we haven't seen any substantial, discernible shift in market share away from the outsourcers (United, Northwest, etc.) and towards the predominantly non-outsourcers (namely AA
Obviously, customers have decided that they have at least reasonable trust the airlines they fly on, and in the trust that these airlines in turn have entrusted with the maintenance organizations they outsource to. I sincerely hope that nothing -- meaning a horrific or catastrophic accident -- comes along to change that perception, but only time will tell.