I think the alt-present scenes are an attempt to show what each character's true nature is, absent any situation where the island draws them more towards either Smokey or Jacob. This didn't occur to me until last night, but it felt like they were saying: Ben is an intellectual guy, searching for meaning in his life, frustrated by powerlessness, and thus capable of Machiavellian manipulations. But inherently, when push comes to shove, he is decent. Which is why in the island-reality he chose, at that crucial moment, to steer away from Smokey and back to Jacob. Sayid, on the other hand, has something inherently violent and evil in him, which is why in the alt-present he killed those guys, and on the island gave himself over to Smokey. Jack is deeply conflicted about his father and has it in him to be angry and conflicted, but in the alt-present he is inherently interested in being a good father himself and breaking the cycle of emotional abuse, so in the island-reality he's on Team Jacob, and so forth.
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I think they are "influencing" what is happening on the island only inasmuch as they show us what these people truly are, in the truest existential sense -- their actual natures are at play in the alt-futures, and those "teetering-between-good-and-evil" natures, I guess you could say, are what Jacob "saw," somehow, and they are what led him to determine that they are "candidates." That moment where Smokey picked up the white rock off the scale and tossed it out the door is more fuel for this theory -- they pick people who are perfectly balanced between "bad" and "good" and bring them to the island as a sort of laboratory to determine which of those forces wins out in the end. So the events that are occurring in the island-present -- the Lost version of "The Stand," where sides are being drawn -- are the "result" (though not really, obviously, in the causal sense) of the alt-futures, wherein we are seeing that left to their own devices, each of these people tilts slightly to one side of the good/evil equation.