This has been called "The Most Obscure Verse in Proverbs". Textual and/or translational uncertainties involve nearly every word in the verse1 and they are to some extent inseparable if we want to end up with a coherent proverb. However, the OP has specifically asked about the translation "archer", so I'll focus on that and only briefly mention some of the available options for the rest.
רַב מְחֽוֹלֵל־כֹּ֑ל וְשֹׂכֵר כְּ֝סִיל וְשֹׂכֵר עֹבְרִים׃
rab mᵉḥôlēl-kōl wᵉśōkēr kᵉsı̂l wᵉśōkēr ʿōbᵉrı̂m
The word of primary interest. The most common meaning is from the root rbb meaning "become great/numerous". The related noun meaning "captain" or "chief"2 is well attested in the Hebrew Bible, most often referring to those in positions of political or administrative power. This certainly explains the JPS choice ("master") and presumably also the KJV ("great God"), although its use as a label for God would be without parallel to my knowledge.
A less common meaning derives from the homonymous root rbb meaning "to shoot". The verb is used only in Ps 18:14. The noun meaning "archer" is used in Jer 50:29, Gen 49:23, and Job 16:13. Although these meager four examples pale in comparison to >400 uses of derivatives of rbb I, above, it may be preferable here given:
- the lack of clear referent for a "captain" in this context, and
- a predicate (see below) that may be more congruent with "archer" as subject.
As written, this poel participle could be from one of two homonymous roots:
- ḥll = "to pierce" or "to wound"
- ḥll = "to begin" (otherwise unattested in the poel, but apparently surmised by KJV etc. to mean "to produce")
Option #1 seems to fit nicely with the subject "archer", which is the basis for modern translations that you cite. This isn't entirely a recent phenomenon: at least as early as 1903, C.H. Toy included this in his compilation of possible meanings (from Bickell, 1891), promptly rejecting it.3 He did not provide his reasoning and apparently found the text too corrupt to admit of a translation.
wᵉśōkēr: "and one who hires".
No emendation or significant re-interpreation is required to arrive at KJV's "rewardeth", although in modern English "hires" better reflects the usual meaning of this (very common) term.
kᵉsı̂l: "a fool"
The noun śōkēr is ostensibly the same participle translated "one who hires" above; however, emendations have been proposed to šikkōr = "drunkard", followed by ESV, NRSV and others.
A participle derived from the root ʿbr, "to pass over", hence "passers-by". By extension, the same root can also mean "overstep, transgress", hence "transgressors".
The NRSV, ESV, and HCSB offer essentially identical translations that accept the emendation to šikkōr ("drunkard") but otherwise follow the MT, accepting the attraction of "archer" as a subject for "wounds":
Like an archer who wounds everybody
is one who hires a passing fool or drunkard. (NRSV)
This seems eminently reasonable to me.
Note: The most deviant translation quoted in the question is NETS, translating the Old Greek. The relationship between the Hebrew and Greek texts of Proverbs in general is uncertain (see the NETS introduction for an overview), and the Greek here may well reflect a different Hebrew Vorlage. I have not seen any translations or commentaries suggesting that Greek offers insight into the original intention of rab. I have thus set this aside for the purpose of this answer, but it may be a worthwhile question to explore separately if interested.
1. The BHS apparatus includes 6 notes for the 7 words in the verse!
2. Whence "rabbi".
3. Toy, C. H. A Critical and Exegetical Commentary on the Book of Proverbs. (Scribner, 1902), pp 475-6.