The stories of the Arabian Nights have had a long and continuing influence on Western art, literature, music, etc. This influence is ably dealt with on other sites. See The Journal of the 1001 Nights and Wikipedia for some pretty good information on Nights influence.
My interest in the Nights rests in the print history. Part of many books is illustration. The Nights offers a plethora of potential subjects for illustration. Despite the mass of potential subject matter, certain subjects are repeatedly chosen by different artists through different times. The setting of the frame story, Shahrázád telling her stories to her sister Dunyázád and Shahryar listening on, is one that has often been the subject of an illustration.
The earliest edition of the Nights, comprising volumes 1 through 4 exist in only a single known copy housed at Princeton University. (Call Number: 2263.2706) The description indicates that there are illustrations. I have not yet had a chance to see these volumes. According to The Arabian Nights Encyclopedia this first edition contained an illustration of Shahrázád and Shahryar as the frontispiece. These illustrations would be the first based on a Nights theme to appear in an English version of the tales. The earliest edition of The Arabian Nights that I have seen is a 1707 edition of volume’s 5 & 6. This I believe to be a first edition in English of those particular stories. That volume contains a single frontispiece illustration, show below. The subject of the illustration is the frame story of Shahrázád, Dunyázád and Shahryar.
Shahrázád, Shahryar and Dunyázád are present in the image. The style of the image is very Western, with Shahrázád and Shahryar resting in a canopied bed in a vaulted room. The setting represents an artistic re-interpretation of the events of the frame story. Here we have Shahrázád apparently telling her stories to Shahryar with Dunyázád as the passive bystander. Shahrázád is completely focussed on Shahryar. Dunyázád has apparently been forgotten. An understandable situation as it is Shahryar that holds Shahrázád’s life. He is the one who must be entrapped by the stories. But, for me, this representation makes the object of her stories too obvious. Shahrázád’s plan needs the Sultan to be unsuspecting. If he were to suspect that Shahrázád was attempting to delay her execution it is much less likely that he would be beguiled by the stories and much less likely that he would be willing to postpone the execution. Never the less, it is quite common to see Shahrázád addressing the Sultan directly with her story telling in images of the event.
I am not very familiar with the early Galland editions. According to The Arabian Nights Encyclopedia the first French edition of Galland was not illustrated. It is not clear when subsequent editions began to be illustrated but later editions most definitely were illustrated. From a 1768 edition, we have the following illustration. The centre panel shows Shahrázád telling her stories. It is very similar to the 1707 version. The setting, location of the actors, even the hand gesture that the Sultan is making and that Shahrázád is making have been reproduced. The later illustration must have been copied from some earlier version.
The surrounding panels illustrate scenes from some of the stories Shahrázád is telling. Clockwise from the top right panel we have: “The Third Kalandar’s Tale,” “The Tale of the Grecian King and the Physician Douban,” “The Merchant and the Genie,” “The Fisherman and the Genie,” and “The Porter and the Three Ladies of Baghdad.”
The artists of both illustrations are unknown.
 Marzolph, Ulrich & Richard van Leeuwen. The Arabian Nights Encyclopedia. Santa Barbara: ABC-CLIO, 2004. Vol. 1, p. 30.