The way things are now, the worst case scenario is that a researcher deserving of the prize dies suddenly before having been recognized. (Of this the starkest example is probably John Stewart Bell, the author of one of the greatest discoveries of theoretical physics, who died of a stroke at 62 before being able to claim the prize that was rightfully his; but lots of slightly less outrageous omissions can be produced.) In order to minimize the probability of things like that happening, the Committee presumably disproportionately favors very old researchers (the reasoning being: X deserves the prize slightly more than Y does, but X is 43 and Y is 97 so we better give the award to Y while we still can). With the possibility of posthumous recognition, that would change. Cases similar to Professor Y would become less "urgent," so more of those scientists would get pushed down the line until they did actually die. And that's a cost, since being recognized while you're alive is certainly better than being recognized when you're dead (though the latter is not worthless; the Nobel prize comes with a considerable amount of money which you can leave to your loved ones). Another consequence would be that relatively younger and more deserving researchers would probably face shorter waiting times between finishing their Nobel-quality work and actually getting the prize.
Overall, I think having posthumous Nobels would be better, but the ultimate cost-benefit analysis depends on lots of details which I don't claim to know. And of course, the possibility of receiving the Nobel prize posthumously should be restricted to those researchers who were alive when the new rule came into effect. It wouldn't really be fair to contemporary scientists to have to all of a sudden face competition from Isaac Newton or David Ricardo.
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