Jaynes was aware of MWI. Jaynes and Everett corresponded with one another, and Jaynes read a short version of Everett's Ph.D. dissertation (in which MWI was first proposed and defended) and wrote a letter commenting on it. You can read the letter here. He seems to have been very impressed by the theory, describing it as "the logical completion of quantum mechanics, in exactly the same sense that relativity was the logical completion of classical theory". Not entirely sure what he meant by that.