If I'm correctly understanding the scenario you described, then you drove a vehicle that was working perfectly well into the dealer for an unrelated matter (the recall), and now the dealer claims the vehicle needs a new and expensive alternator and won't start at all.
If that's the case, I would (politely) discuss with them that the vehicle was dropped off in working order, so if it is now mysteriously not working after being in their hands, they likely did something to it and need to put it back in working order at no cost to me. I mean, I've paraphrased here, but that would be the gist of the conversation, and I would insist that they fix it right away so you can drive it the hell out of there. You've got to do it with honey, not vinegar, but forceful nonetheless.
A great negotiating/influencing tactic is, "I brought the car here and it was working just fine. What can we do to get my car working again without me replacing the alternator?" This puts the burden on them to find solutions, and if it's not a solution you like, you can point out why and just keep repeating the question until you get a satisfactory answer. If you are getting no help, then it's probably time to drop hints about escalating the matter -- to the dealership, to corporate, to the media, the better business bureau, state attorney general -- whatever it takes.