Back when A Realm Reborn was about to launch, the roleplay community learned that, like in 1.X, there would be no official RP servers.
So they got together, and selected one legacy server and one non-legacy server as the unofficial RP servers, Balmung and Gilgamesh.
While the Gilgamesh experiment pretty much crashed and burned for various reasons, Balmung thrived with an ever-growing RP community. As a matter of fact, it thrived too much.
The lack of official status as an RP server caused Balmung to attract both the vast majority of the RP community, and its fair share of non-roleplayers. That's the primary reason why it's so overpopulated.
Afterwards, it kept attracting RP-oriented refugees escaping the dwindling Gilgamesh community and non-roleplayers who simply wanted to play with their friends or wanted a high-population server.
A potential solution to this is to to turn Balmung into an official RP server, amending the mistake made of not giving roleplayers a clear home when the game was launched.
some of you are probably thinking: "wouldn't this make Balmung even more overpopulated?"
yes and no.
Of course this would have worked much better if implemented from the very beginning, but having the server labeled officially as an RP one would probably serve as an encouragement for non roleplayers to move to other servers that aren't officially recognized as dedicated to a playstyle they don't care about.
Most probably, at such an advanced stage, this wouldn't be enough by itself, so Square Enix should probably implement tangible incentives for those who move away. I doubt that what is offered now will be enough to convince any relevant percentage of the population to make the jump. As for what those could be, I let Square Enix to decide, or others to suggest, but it needs to be quite a lot more tempting than a free transfer.
Once the server is established as an RP one officially, and people who don't care for RP start leaking out to other servers, Square Enix should reopen transfers to Balmung for those who want to join the RP community.
Basically everyone would win from this.
Roleplayers on Balmung get official recognition, and on the long run a steady influx of like-minded people from other players.
Non-Roleplayers currently on balmung get to take advantage of tangible incentives to move to a less populated server, where they can invite their friends without worrying of closures.
Roleplayers currently in other servers get to move to a bigger and more thriving RP community of they so wish.
On top of that, Square Enix has every reason to cater to roleplayers. As customers, they're among the steadiest and most profitable. They're more liable to spend money on the mog station in order to get the latest glamors, and they have an additional reason to be attached to their Final Fantasy XIV characters. Regardless of the content Square Enix releases, they always have a reason to stay (and to continue paying the sub), because they effectively self-generate content they enjoy.
Of course no one should feel forced to either leave (non-roleplayers) or join (roleplayers on other servers) Balmung, but giving it an official RP tag would at the very least make it less desirable for a sizable part of its massive population.