Burckhardt was a businessman from Basel, specializing in silk manufacture. According to the site of Basel's Institute of Archaeology, he was one of the richest men in the town, commissioning, for instance, the "Stadtpalais," a residence known as "das Haus zum Kirschgarten" (the house at the cherry orchard), which is now part of Basel's Historisches Museum.
The entry on Burkhardt goes on to say that his mansion housed lots of art, including many plaster casts of works of antiquity, and that he was in contact with such eminences as Goethe. I can't find any mention of him in any of my Goethe reference books, but who knows? I wonder if he is related to Johann Ludwig Burckhardt (1784-1817), a famous Swiss Orientalist who discovered the Roman city of Petra.
It's an indication of the fascination of the Middle East for Europeans that this Burckhardt, during his travels in the Middle East, disguised himself as an Arab (as pictured here), calling himself Sheik Ibrahim ibn Abdallah. His writings -- or perhaps news of his travels in the Middle East -- may have been familiar to Goethe. The scene pictured in the drawing at the top of this post, however, of the young boy handing a New Year's card to his father, would certainly have been one familiar to Goethe. The mother looks like Charlotte von Stein.