One possibility in a fanfiction where Jiraiya is alive could be that he returns during the Fourth Great Ninja War. He could use his vast knowledge and powerful jutsu to turn the tide of the war. Maybe he teams up with Naruto again and helps him master new forms of the Rasengan or other powerful techniques.
In this fanfiction, Boruto might learn some unique ninja techniques from Jiraiya. Jiraiya could see the potential in Boruto and train him in his own unorthodox ways. Maybe Boruto would be introduced to Jiraiya's toad summoning, which could add a whole new dimension to his combat abilities.
Jiraiya could act as a mentor to Boruto. He might correct Boruto's mistakes in combat or give him advice on how to handle difficult situations. Boruto would listen intently, trying to soak up all the knowledge from this legendary ninja. And they could also have some light - hearted moments, like joking about Naruto's clumsy moments when he was young.
Well, Jiraiya being alive could mean a different path for Sasuke. Jiraiya might have been able to convince Sasuke to return to the Leaf Village earlier. He could have used his own experiences and wisdom to show Sasuke that there are other ways to seek revenge and power. And this could have led to a more united and stronger Leaf Village much sooner.
No, Boruto is alive in the manga. The plot hasn't led to his death. However, the manga is ongoing, and anything could happen in the later developments.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he wants to explore his own path independently and test his skills without Jiraiya's constant guidance.