When it comes to negotiations, agreements, decisions and money, people can get emotional and most of it is unrelated to you or your proposal.
You have to be trained to be in control of you and your emotions. You have to be trained to make nothing of it, acknowledge your prospect, and certainly avoid showing any negative emotion yourself. Acknowledge their reaction or whatever experience they may be going through but do not respond with more emotion from your side.
Think of it like this, when a baby starts crying should the parent do the same? I hope you said, “No.” When your prospect starts emoting, he/she is simply expressing some feeling and that should not be given more value than it deserves. Instead of reacting, remain calm, stay interested in them and continue in the negotiations until you get the close. Don’t run from the discomfort, don’t try to hide from it, don’t pretend it isn’t happening and absolutely do not react to it. Stay calm and remain focused on what you’re there to do!