When someone cries easily over anything, what do you call this person?
One-word possibilities include weeper for a person who weeps, crier for a person who cries, and bawler for a person who bawls; see Merriam-Webster.
These words do not always convey weeps, cries, or bawls "easily", but I've heard crier and weeper used in that context. For example, former U.S. Speaker of the House John Boehner often cried in public, and he referred to himself as a crier. The media also referred to him as a crier. (Google boehner cries) Some even dubbed Boehner the "Weeper of the House." (Google boehner weeper)
So crier, weeper, and even bawler seem at least close.
From the most to the least insulting, I would suggest the following list. However some of the later terms are veering into euphemism rather than being strict synonyms.
Such a person could be described as lachrymose, though it's typically used in relatively formal or literary settings, rather than in everyday speech.
lachrymose - Given to shedding tears readily; tearful.
Feelings of sadness and tearfulness can be a result of bereavement or a clinical condition such as major depressive illness. If we exclude clinical conditions and consider a personality trait where minor criticism, insults, hints or the like, may be enough to make a person burst out crying, we could say they are oversensitive or thin-skinned.
One can be oversensitive and still not cry. However, when even minor criticism or negative comments can trigger that reaction, they are certainly oversensitive, touchy or thin-skinned.
If you want an adjective, weepy (mentioned in a comment) works great for this.
If you want something more descriptive, you can say I cry easily. This works really well in real-world contexts; for example, someone has just told you a story about overcoming personal adversity and is surprised to see you crying. While dabbing at your eyes, you can explain:
I cry easily. What a wonderful story!