I have a directory as such:
D:\\Movies
D:\\Movies\\MovieTitle1\\backdrops\\
D:\\Movies\\MovieTitle2\\backdrops\\
D:\\Movies\\MovieTitle3\\backdrops\\
D:\\Movies\
Short answer:
FOR /d /r . %%d IN (backdrops) DO @IF EXIST "%%d" rd /s /q "%%d"
I got my answer from one of the countless answers to the same question on Stack Overflow:
Command line tool to delete folder with a specified name recursively in Windows?
This command is not tested, but I do trust this site enough to post this answer.
As suggested by Alex in a comment, this batch script should be foolproof:
D:
FOR /d /r . %%d IN (backdrops) DO @IF EXIST "%%d" rd /s /q "%%d"