Can an arcane-only caster make use of a Prayer Bead of karma

Yes

Ultimately, Use Magic Device exists, and anyone can activate nearly any magic item. The DC to “emulate a 1st-level class feature,” e.g. the ability to cast divine spells as required for the prayer bead of karma and as possible for many 1st-level classes, is 21:

Emulate a Class Feature: Sometimes you need to use a class feature to activate a magic item. In this case, your effective level in the emulated class equals your Use Magic Device check result minus 20. This skill does not let you actually use the class feature of another class. It just lets you activate items as if you had that class feature. If the class whose feature you are emulating has an alignment requirement, you must meet it, either honestly or by emulating an appropriate alignment with a separate Use Magic Device check (see above).

^{(Use Magic Device skill description)}

The bigger question is whether or not activating a prayer bead of karma does an arcane spellcaster any good—and the answer to that is also yes. The prayer bead of karma simply says that, once activated, the “Wearer casts his spells at +4 caster level. Effect lasts 10 minutes.” Is our wearer casting spells? Then they’re at +4 caster level. Doesn’t say anything about divine vs. arcane here.

Now, the prayer bead of karma, as we already established, requires the ability to cast divine spells. There is a very, very strong trend among many players (including those playing the role of GM) to assume—or want, or prefer—that requirements line up with benefits. If you need divine spells to activate it, then it should only benefit divine spells. This applies to tons and tons of things, and often leads to arguments.

I can’t tell you how to rule at your table. I can tell you that nowhere in the game rules is there even the hint that effects have any necessary or inherent connection to the things you’re required to have or do in order to get that effect. A prestige class that says it progresses “spellcasting” without specifying arcane or divine will advance an arcane class even if the class requires divine spells to enter. A prayer bead of karma will work for a wizard. Those are the rules—the GM can change them, and should iff that would be best for the table, but “officially,” the answer to this question (and many similar ones) has to be “yes.”

About that opening paragraph

The first paragraph for the string of prayer beads describes actually casting a divine spell, and then learning how the prayer beads work. That, arguably, cannot be emulated by Use Magic Device—since you’re not “activating” the item, you’re just triggering a special effect it has. However, it’s also irrelevant: all that does is teach you how the item works. There are plenty of other ways to learn how magic items work—after all, most magic items don’t have any special rules for learning them. So the wizard or whatever can happily just apply identify after detect magic registers the prayer beads as magical, and learn all about how you have to be able to cast divine spells and that you can get +4 to caster level for 10 minutes if you do.

About balance

A prayer bead of karma is worth 20,000 gp, and provides a +4 bonus to caster level. An orange prism ioun stone is worth 30,000 gp, i.e. 50% more, and provides a +1 bonus to caster level, i.e. 75% less. The difference is that the prayer bead of karma only works for 10 minutes, once per day, while the orange prism ioun stone just works, always.

Most magic items in Pathfinder that provide a static bonus scale quadratically with that bonus, that is, b²×c, where b is the bonus and c is some amount of gold. Resistance bonuses to saving throws are b²×1000 gp, enhancement bonuses to ability scores are the same, enhancement bonuses to natural armor are twice that (c = 2000 gp), and so on. It is therefore reasonable to look at the orange prism ioun stone and imagine a +4 version costing not 4× as much, but 16× as much—480,000 gp. That’s 24× the cost of a prayer bead of karma for the same bonus.

On the other hand, “1 day/day” is 144× “10 minutes/day.” Even if you argue—reasonably—that the adventuring day is more like 8 hours, that’s 48× the 10 minutes that a prayer bead of karma gets—twice the 24× factor we found in the cost difference.

So it’s not outrageous to imagine that the prayer bead of karma might be balanced. The bigger issue is that it really depends on how your adventuring day looks to judge its worth—if you can get all your spellcasting done in those 10 minutes, it’s a huge steal, a bonus worth 24× its cost. On the other hand, if you activate it, have it for one fight where you cast a couple spells, and don’t have any other spellcasting to do in those 10 minutes and all your other spells are cast without the bonus, it’s worth a lot, lot less—quite plausibly less, even, than the orange prism ioun stone.

Note, however, that none of the above considered arcane vs. divine—that’s because it doesn’t matter. Clerics and wizards, oracles and sorcerers, these are some of the most powerful classes in the game, simply because of the dominance of spellcasting, and all of them would love a big bonus to caster level. It is very, very difficult, I think, to convincingly argue that either arcane or divine is definitively stronger than the other, or benefits more from this bonus.

Also, not for nothing, but there’s a facet of the prayer bead of karma we haven’t considered—you have to activate it. That takes a standard action,¹ which basically means you should almost never use it mid-combat. That can matter a very great deal, as adventurers don’t always get the luxury of pre-buffing right before a combat. The cleric spell list, however, is better suited to taking advantage here: since it tends to focus a lot on buffs, it’s much easier for clerics or oracles to cast the spells they need during the 10 minutes, and then enjoy the heightened caster level for as long as the buff lasts. Sorcerers and wizards, by contrast, tend to focus more on spells that are cast mid-combat—which means they need to know that a fight is coming up in the next 10 minutes in order to benefit.

All that said, while the prayer bead of karma is no better for arcane spellcasters than it is for divine ones, and by the numbers it is reasonably balanced, it can be abused. My argument here isn’t that the prayer bead of karma is necessarily balanced—only that it being usable by arcane spellcasters doesn’t change that question.

1. _{Magic items with no listed activation type—as is the case for the prayer beads of healing, karma, and summons—default to command word, and command-word items take a standard action to activate.}