In some fanfictions, Minato saves Naruto physically. For example, he might use his teleportation jutsu to appear right in the middle of a fight and whisk Naruto away from danger. In others, he could save Naruto spiritually. He might enter Naruto's mind, like in the fanfic 'Minato's Mind - Saving Mission', and give him the strength and confidence to overcome his inner demons. Sometimes, Minato saves Naruto from a political situation. For instance, in 'Minato and the Hidden Leaf's Politics', he uses his influence and wisdom to protect Naruto from being used as a political pawn.
Well, often in these fanfictions, Minato saves Naruto by using his powerful jutsu. He might use the Rasengan in a new and improved way to defeat an enemy that was about to harm Naruto. In a fanfic like 'Minato's Second Chance', he comes back to life with enhanced powers and uses them to shield Naruto from attacks. Also, Minato can save Naruto by teaching him things. In 'The Lessons of Minato', he imparts his knowledge to Naruto, which helps Naruto save himself in a dangerous situation, which is also a form of saving in a way.
One great 'Minato Saves Naruto' fanfiction is 'A Father's Promise'. In it, Minato manages to break free from the death seal's constraints just in time to save Naruto from a powerful enemy. Another one could be 'Minato's Return', where Minato is brought back to life through some powerful jutsu and he immediately focuses on protecting Naruto. And then there's 'Saving His Son', which shows a very emotional journey of Minato coming back to save Naruto from a life - threatening situation.
Often, Minato could be using his authority as Naruto's father or as a respected figure in the village. He might have made a promise to another clan or village and sees Naruto's marriage as the way to fulfill it. Another possibility is that he believes Naruto is not making good choices in his love life on his own and decides to take matters into his own hands to ensure Naruto has a stable relationship.
The emotional bond would be stronger. Naruto wouldn't have to face the pain of growing up without a father. Minato could be there to cheer him on during his training and be a source of comfort when Naruto faced difficulties, like when he was bullied as a child. He could also help Naruto better control the Nine - Tails' power from the start.
He uses his skills as a ninja to protect Naruto. He sets up special seals around Naruto's crib to keep out any potential threats. Also, Minato might try to use his chakra to boost Naruto's weak body. He would transfer his own life force a little bit at a time to help Naruto fight the sickness, which is a huge sign of his love.
Well, in some fanfictions, Minato might use his powerful jutsu to break through the barriers between the living and the dead and directly reach Naruto. He could also try to influence the people around Naruto to bring him closer to where Minato is, like using his spirit to communicate with Naruto's friends and convince them to help.
Minato's presence can have a huge impact. He can act as a mentor, teaching Naruto things he never knew about his family and heritage. His wisdom and skills can inspire Naruto to reach new heights in his training and ninja development.
Minato might show his protectiveness by always being near Naruto in spirit. For example, he could leave behind some kind of jutsu or power that activates when Naruto is in great danger, like a hidden seal that releases a powerful defense.
One way it could happen is if Naruto somehow gains the ability to travel back in time. Maybe through a powerful jutsu or a unique kekkei genkai. He could then use his knowledge of future events and his combat skills to save his parents from their tragic fate.
Minato would probably be very proud. He might notice Naruto's use of his own jutsu and be impressed by how Naruto has developed his skills. Kushina would be more emotional, maybe crying at the sight of Naruto growing up so well.
Well, Minato might constantly be by Naruto's side in the fanfiction. He could prevent Naruto from taking on any dangerous missions, even if Naruto is capable. For example, he may interfere when Naruto tries to join a risky battle against a powerful enemy, saying it's too dangerous for him.