I agree with Softly Spoken.
There are generally people who cry because they can't help it, and people who do it on purpose as part of their personal pity party. It can be hard to tell them apart. But luckily that doesn't matter in how you can deal with them.
Be empathetic and console them if possible, but do not recant or change your mind about the thing that made them cry. They are an adult and can deal with it. If you tell someone that you don't need their help and they burst into tears, focus on making them feel better, but do not change your mind. Focus on the fact that you don't like seeing them upset, not what made them upset. No matter what the reason, a person cannot deal with things well while crying. Help them feel better first before proceeding. Do not fall into the trap of discussing things with an upset or even hysterical person. It will not end well. Be a broken record with "we'd better talk about this when you're feeling better" or something similar and do not engage.
And, if you are busy, give them an opportunity to leave (so as not to embarrass themselves) while you continue what you were busy with. "Oh, I see you're upset. Why don't you get some fresh air outside to help feel better while I finish up here?" This also works for repeat offenders.
People who cry on purpose do so to make you feel/look bad. If you help calm them and are empathetic, you will not look bad. And as long as you do not confuse making them feel better with doing what they want, you will not be manipulated.
I had a student in year 9 who would literally burst into tears whenever he wanted something that was denied to him. It really surprised me the first time. I held him back after class to give him a detention for cheating in a test, and he started sobbing and crying! I did what I could to make him feel better, feeling very awkward, and as soon as he seemed to be calming down, he asked, "but do I still have to do the detention?" I responded with something like, "yes, since the fact that you cheated still stands." He immediately started crying again and I then knew for certain that it was a ruse. I made it perfectly clear that being upset about something does not absolve him of his actions after I had calmed him again.