An ambassador's entourage was more about appearance than protection, so going alone added little risk. Besides, she liked what going alone said about her. She was Zelda, bearer of the Triforce of Wisdom, Sage of Time, and she needed no one. Her kingdom needed the aid of the Zora people, but that was beside the point.
Also, it was probably for the best to leave behind anyone she knew, because she would have to adopt Zora culture for a time, and the regal gowns she usually wore would not do. Neither would the skintight garb of the Sheikah that she had worn throughout her exile, because the Zora's domain was filled with water, and whatever she wore would have to handle getting wet.
In fact, if she were to completely embrace Zora culture, she would have to go nude. The mere thought made her heart race, but the Zoras did not share her people's taboos against nudity. To them, nudity simply wasn't sexual. Actually, they were more prudish about sex than Hylians, but to them, wearing clothes was simply impractical for swimming.
Alternatively, she could compromise. She had a bikini that was suitable for swimming (barely) and nothing else, and while it wouldn't have the same effect as total nudity, then at least if another non-Zora passed through and saw her, she be able to live it down.
