A question that I was asked a few days ago by a family friend, but i was unable to provide a good answer, as it is far from my area of research. There were a couple of questions that were posed:
What was the size of this organisation? Did they have much sway on Spartan politics? What was the extent of the killings of the helots?
Hopefully someone can answer these questions!
It's worth noting that the lurid details of the krypteia do not show up until Plutarch, half a millennium after what he was describing. It is not described by Xenophon or Aristotle. Many such as myself question the extent to which it is reality or later imagination.
It's only Plutarch that tells us the popular stories about the krypteia going around hiding by day (hence krypt-, "hide") and murdering helots at night; but Plutarch is extremely unreliable on early/Classical Sparta. His testimony has little value. It's best to treat his account as mostly fiction.
The only earlier evidence we have is that Plato also describes the krypteia briefly in his Laws. He states that it was simply a form of military training under very harsh conditions.
One thing that may be salvageable from Plutarch, though, is when he tells us in a different place (Life of Kleomenes 28) that the krypteia fought in a battle in 222 BCE. That implies that they were a military unit, and not a training regimen that was given to all Spartiates universally.
Please keep in mind that we know very little about the krypteia, and much of what we know was written after their existence and by outsiders who had a major axe to grind with Sparta. I have a degree in classics, but have studied ancient military history as a hobby.
The size of the organization was probably very small, as it was only a small selection of an already small pool of recruits. It is said(Plutarch?) that Spartiates who took part in the krypteia were the ones to achieve the highest ranks in the Spartan military later in life.
This would make sense if they served as secret police or carried out executions. These men would be the ones that understood the whole picture of what the Spartan lifestyle caused on a regional scale, and how the Spartan way of life was able to survive.
I disagree with some historians that claim that they roamed about the countryside murdering helots for "training". The helots in general were a vital part of spartan society, and were generally well treated as far as "slaves" go until the messenian revolts and the beginning of the decline of Sparta. I do think there would have been executions of helot rabble rousers though, as nothing destabilized the Spartan state worse than helot uprisings.
It's also possible that it was just additional fieldcraft training for gifted Spartiates. They were known for stealth and secrecy and roaming the countryside with no provisions. It would not be a surprise if it was just additional fieldcraft training for reconnaissance missions as part of the army.
In summation, I think it would be accurate to depict the krypteia as a small branch of secret police, with possibly some assassination duties of helot revolutionaries. They possibly performed a scouting role within the army as well.
There are several books written about them, but I really find most of them very slanted and opinionated. This is probably because there is so little information to go on, so the authors fill in the gaps to support whatever theories they have about them.
Sorry that I cannot provide a more precise answer. But the krypteia isn't a very widely documented organization.