He may show a rare moment of kindness. Given his own past traumas and experiences, he might empathize with Hermione on some level. He could offer her some words of wisdom or even some form of magical assistance to help her get better.
Snape could be hesitant at first. He might not know exactly how to handle the situation. But as he realizes the severity of Hermione's state, he could become more involved. He might start investigating what caused her to be in this 'broken' state, using his skills as a Potions master and his knowledge of the dark arts.
He might be initially startled. Snape is used to seeing Hermione as a strong - willed and confident student. So seeing her crying could catch him off - guard. But then he might quickly put on his usual cold facade and ask her curtly what's wrong.
Snape might be suspicious at first. He might think it's some sort of trick or part of a bigger plot. However, as he observes Hermione more closely and realizes the genuineness of her situation, his attitude could gradually change to one of reluctant assistance. He might even offer some words of wisdom, albeit in his typical sarcastic way, to help Hermione regain her strength.
He might immediately go into a protective mode. He could use his potions knowledge to quickly make a healing draught for her. After all, Snape is very skilled in potions, and Hermione being hurt would likely trigger his instinct to help.
Snape would likely be shocked and then enraged. Harry is supposed to be the hero of the wizarding world, and seeing him drunk would be a huge disappointment. He would most likely drag Harry to his office, lecture him about propriety and the expectations placed on him as a wizard, and probably assign some form of punishment like a long detention. But as he observes Harry more closely during this time, he might start to wonder what drove Harry to this and if there are deeper issues that need to be addressed.
One possible reason could be the stress from her studies. Hermione is always under a great deal of pressure to excel academically. Maybe she reached a breaking point due to the overwhelming workload. Another reason might be some kind of conflict with her friends. For example, if she had a big fight with Harry or Ron over something serious, it could leave her feeling broken.
He is often very suspicious. He might immediately think it's some kind of trick by the Marauders or their descendants. He'll likely try to figure out how to use it to his advantage against those he dislikes.
She might be initially defensive. Hermione is a proud and independent character. So, when Snape finds out about her arm, she could be worried that he'll use it against her in some way, like taking away house points or using it as an excuse to give her extra work.
Snape might initially be taken aback. He's not used to students standing up to him so forcefully. He could glare at Hermione with his cold, dark eyes, trying to intimidate her into silence.
He could be very surprised at first. Maybe he's just out on a normal day in the forest and suddenly comes across a baby. His immediate reaction might be to look around for any sign of the baby's parents.
Snape might initially be shocked. He's not one to show much emotion openly, but internally he could be very disturbed. He might start investigating what happened to Hermione immediately, using his skills in potions and magic to try and find a cure or at least understand the cause.