This answer is going to read like a combination of a few other answers, but really I'm just going to provide a partial summary of Turn A Gundam, which has a background somewhat similar to the setting you propose.
In the series, the Earth and everything on it was nearly destroyed by war, setting Earth culture back thousands of years. Meanwhile, on the moon, humans continued to get by more or less just fine. The culture of the moon is (not entirely) utopian, and citizens appear to be loosely separated into castes.
The series deals with warrior/soldier castes in a few different ways:
* Royal Guard - Protectors of the regency, their purpose in life is to ensure the safety of those in power, because sometimes rebels just happen (politics, boredom, desire to resettle the Earth, etc.)
* Cryogenically Preserved Soldiers - As part of the setting of Turn A, certain members of society are occasionally cryogenically preserved and resuscitated at regular intervals, including nobility, those with (presently incurable) illnesses, and soldiers with extensive experience and ability (who are loyal to the regime/society). The soldiers don't necessarily like "going back to sleep" but their loyalty compels them to follow orders, so if their presence is deemed unnecessary or dangerous they'll still be willing to go back into storage.
* Legit Born-and-Bred Warriors - These people are raised with the promise of war, that some day they would be invaluable to the survival of their culture. In the mean time, they train via simulations (VR, games, whatever) and physical training exercises. One thing to note is that everyone shown to be a part of this faction is particularly hot-blooded and violent. 164
Whether this is nature or nurture is (IIRC) not made explicit.
* Civilian army - This group is the largest, but also possibly the weakest. Completely voluntary, their training doesn't seem anywhere near as extreme or thorough as the (much smaller) legit warrior caste, but they do seem fairly capable as soldiers, i.e. following orders and persevering in the face of unexpected difficulties. This army may or may not be a standing army, instead coming about mainly to facilitate the invasion of Earth. As such, I hesitate to consider this a "caste" and more just a profession (as it is now, more or less).
One thing the series also makes clear is that peaceful utopian societies don't produce a whole lot of exceptional soldiers/warriors, the moon culture's primary advantage over the Earth's being technology, not guts nor even strategy. Training, simulations, and psychosis help even the playing field, but against a foe that has spent centuries struggling and warring, it might not be enough.
TL;DR there are a lot of options, and there's nothing wrong with diversification. 164