This is my favorite translation of The Tale of Genji. I feel though, that I must clarify my opinion on that, for it varies, depending on what kind of reader you are.
If you are a casual reader who wants to read this book simply out of interest, or a desire to build upon your stockpiled erudition, or you're an anime nerd who wants to learn more of Japan than just Gundam Wing, then this is NOT the translation for you. Read the Seidensticker translation.
If you love Japanese literature, or just literature in general, then read this one. If you don't mind spending a bit of extra time to comprehend everything, then read this one.
The difference between the two is this: the Seidensticker translation is very easy to understand and actually calls each character by their respective names. Each sentence tends to be very simple, as in "Genji was sad. He went to the shrine at Izu. There he met the Priestess."(not a quote) As for the Royall translation, he writes more poetically, so to capture the feeling of the text, but he also calls each character by their proffesional title, rather than their name, which changes depending on promotions or demotions, which occurs often. For example, Genji will be called the Grand Counselor in one chapter, and then called the Palace Minister in the next. Fortunately, Royall offers us a little 'Dramatis Personae' at the beginning of each chapter so as long as we read that before starting each new chapter, everything should be alright.
So, in summation, this is the best Genji translation as far as beauty in prose goes, but it's harder to understand.