How often do you know that you are absolutely right and that your customer is absolutely wrong? It must take all of the patience and tactfulness in the world for a travel advisor to deal with such a challenge.
Of course, some "wrong" clients aren't worth keeping. If they swear up and down they have passports and call you from the airport as they can't get on the plane to Europe because they had no idea they needed a passport, it's not worth your time trying to send them on a follow-up trip. The customer who argues with you on price, falsely claiming you cited a very low quote for a vacation, is definitely not worth saving. All you can do in these cases is extract yourself gracefully.
Things are not always so clear cut, however, when social media is involved. I belong to a Facebook group whose members sell their crafts online on Etsy. One seller was lamenting how poorly she dealt with customer service in the early days of her shop. She cited a customer who wanted to return an item. The seller recalled how she wrote several e-mails arguing with the woman as to why she would not take the item back. She had every reason in the world for not accepting this return and she was right on every level. The result? The customer wrote a scathing review, focusing on how the seller was negligent in addressing the buyer's concerns. That's an ouch. That very public, negative review hurt sales for several weeks and the seller definitely wished she had handled things differently. It was most certainly a case of "Win the argument, lose the battle."