It's good question. Based on the style, my guess would be Jim Roslof, who was art director at the time (1983).

I checked Art & Arcana, and while it mentions the design, it doesn't say who is responsible:

"Famously, the trade dress of Dungeons & Dragons this year, both in the Basic and Advanced versions, began to form its ampersands into the shape of a dragon breathing fire, proving further visual distinction between TSR's D&D and non-D&D properties" (pg 151).

Perhaps Jon (@increment) knows?

Ok, basically no one knows it. Apologies for Frank, he does not know that, full stop. We should simply ask to either Elmore or Easley. Is there anyone in touch with these fellows?

aia wrote in Question about D&D logo:if it was in the first part of the year

I remember the Monster Manual II appearing in stores at the beginning of 1983 with the new design.