1 From my What is Sacrifice?, slightly expanded
Khernips refers to water used for washing the hands (Pollux, Onomasticon 2.149), especially before a meal or before sacrifice. At sacrifices, there were bowls or vessels with water to be used as khernips, which could have been passed around or poured out over people’s hands.
Among modern Hellenic polytheists, it has come to be thought that even in household practice, khernips must be somehow specially prepared, because “in sacrifices (hieropoiiai), they used to plunge a fire-brand into the khernips and sprinkle it around the altar” (Hesychius δ 155). But to my knowledge, this is only attested for sacrifices at an outdoor altar, only for classical Athens, and quite sparsely at that (namely in Euripides, Heracles 925f and Aristophanes, Peace 959). It is only one scholium that gives the underlying logic: “They thought that by plunging fire into water, they could purify it, because fire is purificatory of everything, as Euripides writes in the Heracles: ‘The son of Alcmena was bringing the torch in his right hand, to dip it into the khernips’.” (Scholium on Aristophanes, Peace 959). Apparently this ceremony already required explanation to readers of the Athenian classics* in later antiquity, because it had ceased to be observed, and was not found in earlier authorities like Homer.
In short, the use of khernips in household cult at root simply means that you should wash your hands. While more complicated practices are venerable, they are far from necessary, and not at all uniform; see Purification by Water.
Note
* The entry in Hesychius is from an ancient commentary on Euripides, Heracles 925f, as the other note is a scholium on Aristophanes, Peace 959.
2 Verbal Formulae
Instructions for making khernips that can be found online also include certain words to be spoken: firstly, kherníptomai or “I am washing my hands”; then either hekás, hekàs éste bébēloi (“Be ye far off, far off, ye uninitiated!”) or apó, apó, kakodaímones (which is supposed to mean “Away, away, evil spirits!”, but does not). There is no harm in saying something along these lines, even if there is no ancient precedent for it, but the phrases are written/adapted to this purpose by someone with a loose grasp of Ancient Greek.*
Hekàs hekàs éste is not great Greek; apó, apó is nonsense. Bébēloi refers to uninitiated persons, and the unitiated are sent away at the beginning of secret gatherings, not when one is alone in a room making some holy water. Finally, kakodaímones in something like 99.9% of its usage, means “people with bad luck”, not “evil daemons”.
So, while Kherníptomai (Χερνίπτομαι) is perfectly good Greek, something like Thýraze ponēroì daímones (Θύραζε πονηροὶ δαίμονες), “Out, wicked daemons!”, would be preferable to the other incantations (cf. Suda θ 598).
Note
* Apó, apó, kakodaímones in particular appears to be a half-successful attempt to correct the even less grammatical call Apo pantos kakodaimonos, found in Aleister Crowley’s Star Ruby or Liber XXV. The other chant, hekás, hekàs éste bébēloi has been claimed to be from the Eleusinian mysteries, but that is empty speculation. It is a modern mixture of the Latin procul o procul este profani (Vergil, Aeneid 6.258; in Servius’ commentary profani is glossed as bébēloi) and the Greek phrase which probably inspired it, hekàs hekàs hóstis[!] alitrós (Callimachus, Hymn 2.2. As noted above, such a proclamation appropriate to the beginning of an exclusive or sectretive communal ritual, not to making khernips, or banishing daemons in general. Thanks to glamrocktrash and Chelydoreus for most of this information.
SARTRIX 