At low levels, we tend to see Explorers using arbalests for the d8 damage. In the mid-levels, the party starts finding magic bows and the arbalest’s low rate of fire starts limiting the explorer’s ability to cleave, and some of them switch over to bows (the only proficiency in ACKS that cares about bows vs crossbows is Weapon Focus, and Weapon Focus is not typically a priority pick for a ranged combatant at low levels compared to eg Precise Shooting or Skirmishing). Magic composite bows are probably stronger than arbalests at high levels in the presence of Fighter Damage Bonus (which gradually decreases the importance of the arbalest’s +1 expected damage), but at low levels arbalest is very good at one-shotting 1HD monsters up to our explorer’s cleave cap. A 1d6-damage bow has a 43.8% chance to kill a 1HD monster. A 1d6+1 bow-explorer or 1d8+0 arbalest-peasant has a 56.3% chance to kill a 1HD monster. A 1d8+1 arbalest-explorer has a 67% chance to kill a 1HD monster. So at low levels there is a noticeable difference in killing power against weak foes.
Arbalest might remain a good weapon choice for a high-level Assassin or Thief with Sniping, because they benefit disproportionately from the higher damage die when it’s multiplied by backstab. An Explorer or Barbarian could take Ambushing+Sniping+arbalest to leverage this too, and both would be able to make good use of it (the barbarian because of Naturally Stealthy, and the explorer because of the 3+ hide in wilderness), but it wouldn’t scale as well into the high levels as it does for Assassin, and it’s a pretty heavy investment of class proficiencies.
Besides limits of cleaving riding is another thing to remember when riding mounts you can't use long bows or Arbalest (assumingly). Only short or composite bows and assumingly only with the riding proficiency. If everyone else is melee you will really want to pick up precise shooting to allow/remove the penalty to shooting into melee. If all you want to do is be a ranged fighter I would suggest Barbarian as you can get precise shooting to start with and lower that penalty by picking it again and you get more health on average.
They have a similar selection of proficiencies to choose from. So the real question if it is just down to the two of them is do you want more combat ability (More HP and Reroll mortal wounds) or out of combat ability (Hiding, Evading, and Navigation).
Last is presentation and ease of travel. An Arbalest and long bow are unwieldy but effective weapons. They weren’t made to hunt with so dropping one down on a bar table or just carrying it around unslung could be cause for trouble.
Oh and Ammo! Normal arrows cost 1 GP a pack and bolts 2, which doesn’t normally matter but in some markets you will have trouble refiling unless you only buy silvered arrows/bolts which would put them in the same category. Picking up a backup weapon is also cheaper when you stick with bows or if you need to scrounge for ammo more people will have bows than crossbows.
I think I’d probaby allow mounted arbalest use; there’s no mention of it not being usable mounted in its weapon description like there is in the longbow’s. The only reason to use a longbow over a composite bow is if you don’t have the money for a composite bow, and composite is still cheap enough to afford with your starting money.
That is an interesting point about barbarian opening with precise shooting, but it’s worth noting that as an Explorer you can shift ability score points into Dexterity because it’s a prime requisite (per ACKS core page 17), whereas as a Barbarian you cannot. This makes it relatively easy to pick up an 18 Dex as an explorer, while as a barbarian you will need more luck. That additional ranged to-hit (nevermind initiative and AC) from prime-req boosted Dex, plus the explorer’s +1 to-hit with ranged weapons, can make taking a second rank of Precise Shooting unnecessary. The explorer Hide in Wilderness ability is also very, very strong. A well-played explorer can single-handedly win many wilderness engagements, while barbarians are stronger in the low dungeon levels (also worth noting that Precise Shooting is primarily necessary in the low dungeoneering levels; it remains useful in “single big monster” fights at high levels, but overall I don’t think taking it twice is a great long-term decision).