Which is Correct?

The latter. In chess, the king is never entirely "safe"; otherwise there would be no point to playing the game.

If the sentence stands alone by itself, then only "safe" makes sense.

safer = is a comparison. ...make your king safer (than doing - some other move). The "than" part needs to be either stated or understood from the context of what has already been said.

Now, many people might say "safer", even if that's the only sentence said, but that would technically be incorrect without have made or implied a comparison. Safer than what?

On another note, making the king safe is one possible goal of castling, and it's not the usual one. The usual goal is to move two pieces (king and rook) at the same time. That allows to develop that rook, while putting the king in the corner, so it is easier to protect (than leaving it in the center). If you can castle and your opponent either can't or choose not to castle, then you usually have an advantage, because your pieces are better developed.

Safer. In chess, the king is never completely safe so long as the game continues. Castling is sometimes a good idea, and sometimes not. The whole point of the game is to attack the opponent's king while protecting your own, but the king is always vulnerable because of his limited range of movement.

English is not my native language, but from me, the first phrase just says that castle is needed for keep the king safe.

The second phrase suggests an improvement, for keep the king safer.

safe