I know that, at least when they were small, the children had very limited knowledge of Asian cultures and were actually quite prejudiced against them. I wrote the following in an old post but will re-post it here as it is relevant to the topic:
"I have read accounts that in their early childhoods they were prejudiced towards Asians, referring to them as "monkeys," but this seems to have been a product of their resentment towards them during the Russo-Japanese War, as well as them being a product of their times, when people were very racist towards Asians, Jews, and anyone else who was different in race or religion.
In her book, "Six Years at the Russian Court," Miss Eagar recounts a few different times when the little Grand Duchesses displayed prejudiced streaks:
1) Tatiana: This was regarding a visit from the young Prince of Siam to the court: "The Empress came to the Grand Duchess Tatiana, 'Come, shake hands with this gentleman, Tatiana.' She laughed, and said, 'That is not a gentleman, mama; that's only a monkey.' The Empress, covered with confusion, said, 'You are a monkey yourself, Tatiana,' but the prince laughed heartily."
2) Olga: During the Russo-Japanese War: "Olga was working very diligently one day and said to me, 'I hope the Russian soldiers will kill all the Japanese; not leave even one alive.'" After an explanation from her nurse that there were many innocent little children and women in Japan who did not deserve to be killed, that they also had an Emperor, and various other sympathetic points, little Olga said, "I did not know that the Japs were people like ourselves. I thought that they were only like monkeys." Fortunately, Miss Eager tells us that "She never said again anything about being pleased to hear of the deaths of the Japanese."
3) Maria: This occured when the little girl saw a picture of the baby children of the Crown Prince of Japan: "...and with a look of hatred coming into her sweet little face Marie slapped the picture with her open hand. 'Horrid little people,' said she; 'they came and destroyed our poor ships and drowned our sailors.' I explained to her that it was not these little children, who were only babies younger than Anastasie. So she said, 'Yes; those little babies did it. Mama told me the Japs were all only little people.'"
So at least as very little children they could be considered prejudiced, but it seems that it was mostly due to the result of the war they experienced so early on. These kinds of comments were of course also repeated about the Japanese during WWII. And it was also a product of hearing what their parents and other people said, as since everyone knows children absorb everything. They were a product of their times. But I would like to think that they grew out of that as, like Olga, they all eventually realized that we were all just human beings."