First, it's not a complex sentence. It's a simple sentence. "It" is the subject and "broke down" is the simple predicate. "with the crossover appeal of its music" is a prepositional phrase modifying "broke down." The phrase tells how it "broke down." The rule in this case is
- Use commas to set off phrases at the end of the sentence that refer back to the beginning or middle of the sentence. Such phrases are free modifiers that can be placed anywhere in the sentence without causing confusion. (If the placement of the modifier causes confusion, then it is not "free" and must remain "bound" to the word it modifies.There's the rub: Does the phrase cause confusion by use of the comma?) https://owl.english.purdue.edu/owl/owlprint/607/
The prepositional phrase could have been placed at the beginning of the sentence, where it would also be separated by a comma.
With the crossover appeal of its music, it [the blues, let's say] also broke down racial prejudice.
It [the blues, let's say], with the crossover appeal of its music, also broke down racial prejudice.
However, if it were a complex sentence such as "It also broke down racial prejudice when the music migrated to popular radio stations," the dependent clause (when the music migrated to popular radio stations) wouldn't be separated from the main clause, so you're right there.
In reality, you'll see sentences like yours written without the comma.