Snape, being a skilled wizard, could use magic to protect Draco. He might cast protective spells around Draco when he's alone or in vulnerable situations. Maybe he uses a spell that alerts him whenever Draco is in danger, so he can rush to his aid immediately.
Well, perhaps Snape uses his position as a teacher to step in. He could have been observing the situation secretly and when Draco was about to do something harmful to Hermione, Snape intervened with a firm command or some magic spell to stop Draco in his tracks.
Snape would probably be very stern with Draco. He might give him a severe scolding and deduct points from Slytherin. Snape is a strict teacher and he won't tolerate such behavior from his own house students.
Another motive could be related to his complex relationship with Dumbledore. Dumbledore might have asked Snape to look out for Draco, and Snape, being obedient to Dumbledore in many ways, would do so to fulfill his request. Also, Snape might believe that protecting Draco is part of a bigger plan to maintain the balance in the wizarding world.
Well, it could be that Draco, in his pure - blood pride and influenced by his family's views, targets Hermione. Maybe he uses some dark magic or just plain bullying. Snape, despite his often cold exterior, might save her because he secretly admires Hermione's intelligence or perhaps he has a deeper reason related to his own past with the Gryffindors or his hidden loyalties. For example, he could see a bit of himself in Hermione's determination to succeed in a world that is often against her.
Snape could also punish Draco by deducting house points. Since Draco is in Slytherin, losing house points is a big blow to his pride and to the whole Slytherin house. This would make Draco the target of his housemates' ire, adding to the punishment's severity.
In fanfiction, Snape saving Harry often showcases a complex side of Snape's character. It might be a redemption arc for him, as in the original series, Snape had a very ambiguous relationship with Harry. In these fanfictions, it could be due to his love for Lily that he finally decides to fully protect Harry, which is a very interesting take on his character.
One possible aspect to analyze is Snape's complex character. In many 'Snape Saves Harry' fanfics, it delves deeper into his motives. Snape has always had a love - hate relationship with Harry. In these fanfics, his act of saving Harry might be due to his lingering love for Lily. It shows a different side of Snape, one that is more heroic and self - sacrificing than what we see in the original series at times. This also allows for exploring different plotlines and character development arcs for both Snape and Harry.
I also like 'When Snape Becomes Harry's Guardian'. In this fanfic, Snape saves Harry from a dangerous situation early on, and then takes on a sort of guardianship role. It has some really interesting interactions between the two characters as they learn to live with each other.