My first thought of course was to laugh at the concept.  But are we absolutely sure that Caesar would wipe out music, destroy all radios like Al Chestbreach? It's easy to believe that he's against all technology, but that's not precisely true.  

But as far as I know, no one, ever, in all human history, has wiped out all music.  I'm not even sure anyone has tried, not even the craziest wrong-headed emperors.  

One could suggest he'd have new music made that would glorify him, but there's something broken about the people of the Fallout universe -- they aren't good at new music.  They just aren't, for some reason.  It's by now sort of baked into the world.  

Music is technically "useless" I suppose, although news propaganda might be useful for broadcasting the Legion's critiques of the NCR, some of which aren't completely baseless.

I really think Caesar was damn fool for completely destroying Nipton, Searchlight, Nelson, etc.  One dialogue path actually has Lanius understanding the truth of that.

I bet a modded DJ could put together some really interesting propaganda without even having to lie.

I think Ceaser would use it as propaganda like Eden did with the Enclave. Especially during War time, would be used to demoralize the NCR and be a way to spread fear and get tribes to surrender and join the Legion.

This is actually a really cool idea for a mod.

Because we're on the Legion frontier, it could be like a Hanoi Hannah thing. Where you can pick up the "Legion Broadcast" anywhere and it's just constant stuff about how the soldiers are so far away from home, how they'll die out in the middle of the desert with no one to mourn them, what the Legion will do to soldiers if they catch them, what will happen to people who defy the Legion, and reasons for why people should support the Legion, like the increased security, how the NCR is a failed nation, how only unified strength under one ruler can bring about peace. Stuff like that. Maybe draw upon Pax Romana fascist propaganda to really drive home the Legion's ethos. "We are strong, you are weak. To be against us is to be degenerate, dissolute and profligate."

This could be really fun. I'd voice act

tl;dr my long winded ramble goes on.

(Edit: Achilles of Atlanta), in Caesar's New Regime, says somethings that I like, and some things that I think aren't accurate about Caesar's Legion (I don't think Caesar is all about following self interest at all.)  
But he points out how the NCR is re-creating the old world and thereby the blueprint by which it was destroyed in the first place.  That the wasteland is proof that democracy failed.  Whether or not we agree with the philosophy, it's interesting propaganda.  I shudder to think what president Kimball would do with a hundred nukes.  
Combined with select lines inspired directly by Caesar, there's so much propaganda possibilities.  Caesar's legion was not sold properly in the Mojave, when all we can see are all the towns that they burned down foolishly and brutally.  They don't appear to be conquerors, just crazed murderers.  They need some propaganda.
But my biggest question is trying to decide Caesar's view on music.  He says the individual only has as much value as his utility to the state.  Would such a person bother with music at all, seeing it as nothing but a trivial waste of time?  Or would he recognize the possibility of it to motivate hearts?

He seems to be against drugs and alcohol (even stims I guess.)  I take it he'd be against gambling too (which means he'd destroy Vegas' economy and it would end up a ruin like everything else.)  But that doesn't mean he'd have to ban things like Caravan, Poker, Chess (pretend they probably exist in fallout despite not being programmed in the game.)  Those require some intelligence and strategy,and can be played without gambling, and so probably still have some utility.  People play card games in real life even in places were gambling isn't legal -- but I don't know anyone who uses a slot machine that doesn't pay out.  
If Caesar is serious about building a new, unified culture, a source of entertainment and art would probably be central to the synthesis he seems to think will happen after taking Vegas. Radio would be the first step, I suspect.
Edit:
A roman music link to youtube for the lols:  
https://www.youtube.com/watch?v=rgRmnmyNKaU