The baby could be the child of a fallen ninja. Maybe the ninja was an ally of Naruto's and before dying, entrusted Naruto with the baby. This would add an emotional depth to the story as Naruto would feel a great sense of responsibility to raise the baby well.
One possible character could be Jiraiya. He was Naruto's mentor later in life, but in a fanfiction, he could have taken Naruto as a baby. He had a close relationship with Naruto's father, so it would make sense for him to take Naruto under his wing from the start. Another could be Tsunade. As one of the Sannin, she has the power and the means to protect Naruto. Plus, she might have seen it as her duty to take care of the son of her old teammate.
Naruto uses his ninja skills. He's always on high alert, using his enhanced senses to detect any potential danger. His quick reflexes mean he can react in an instant if anything threatens the baby.
Sasuke could influence baby Naruto's growth by teaching him discipline. Naruto as a baby would be impressionable, and Sasuke's strict ways could instill a sense of self - control in him early on.
Well, it's mainly baby Sasuke and baby Naruto. But sometimes other characters from the Naruto universe might make an appearance in the fanfiction. For instance, Iruka Sensei could be the one looking after them, or Sakura as a baby might be in the background. However, the spotlight is mostly on baby Sasuke and baby Naruto as they are the ones around which the whole concept of this fanfiction revolves.
It could be Hinata. Since they are in a relationship in the original series, it would be a natural choice. Hinata would be a loving and caring mother, and her gentle nature would be a great influence on the baby.
Well, he might have shown superhuman strength, like being able to break things that normal babies couldn't. Maybe he could also sense danger in an uncanny way, like a sixth sense.
Kushina would take care of baby Naruto very tenderly. She would make sure he is well - fed before sleep, perhaps by breastfeeding him or giving him a warm bottle. She'd also check his diaper and change it if needed.
Some of the more traditional ninja elders might oppose Naruto's decision. They could think that Naruto, as a busy ninja, doesn't have the time to take proper care of a baby. They might be worried that it could affect his performance in important missions.
Kakashi would approach Naruto's pranks with a mix of discipline and love. For instance, if Naruto tries to scribble on the walls, Kakashi would stop him gently. He'd then sit Naruto down and show him where he can draw instead, like on a piece of paper. He'd also tell Naruto stories about how important it is to respect their home. When Naruto does something really naughty, like trying to pull the cat's tail, Kakashi would firmly say 'No' but then immediately follow it up with a hug. He wants Naruto to learn right from wrong but also wants to make sure Naruto knows he is loved no matter what.