One possibility is that they gain a deeper understanding of Luffy's motives. For example, they might see his memories of making the promise to Shanks and realize how important it is for him to become the Pirate King. This could strengthen their resolve to support him even more.
Luffy might be excited to have a new member, as he always is about making new friends. But he's also not stupid, he would keep an eye on Kuro. Zoro would be very skeptical. He doesn't trust easily, especially someone with Kuro's reputation. Nami would probably try to figure out how she can benefit from Kuro's skills, while also being cautious not to be double - crossed.
The relationships will likely become more profound. For instance, Zoro might see Luffy's memories of believing in his strength even when others doubted, which could make Zoro more determined to protect Luffy at all costs.
It could be Chopper. Since Chopper is a doctor and Reiju has some medical knowledge, they might bond over sharing medical techniques and treating the crew's injuries. Also, Chopper's kind - hearted nature would make him a great friend for Reiju who might be used to a more cut - throat family environment.
If Naruto was raised by the Straw Hats, he'd probably be a very different person. He might be more carefree like Luffy. For example, he'd be used to the wacky adventures on the Grand Line. His ninja skills could blend with the pirates' combat styles, creating some really unique fighting techniques.
Nami would probably be devastated. She has been with Luffy through so much, and his death would be a huge blow. She might become more withdrawn and serious, as Luffy was often the one to bring out her lighter side.
Zoro would be deeply affected. He has sworn to follow Luffy and his death would be a huge blow. He might feel like he failed in his duty to protect the captain. He would likely become more withdrawn at first, but then focus on getting stronger to avenge Luffy if there was someone to blame for his death.
Well, as of the latest manga chapters, Yamato has indeed become a part of the Straw Hats. This decision brings new dynamics and adventures to the crew.