It's more likely to have a positive effect on Costco rather than none at all: if all minimum-wage workers have more money in their pocket, they're more likely to buy a CostCo membership, or spend more if they have one already.
That's highly doubtful as Costco is primarily used by middle income earners, in one of the paradoxical effects of wealth where having a bit more allows one to save more. Walmart would be far more likely to benefit, but we know that Walmart isn't exactly for it.
People are really stretching to try to make Costco out to be bad guys in this.
I didn't mean to imply that there was anything sinister with Costco's advocacy; it simply seems like an instance where the interests of Costco and minimum wage workers (possibly) align.
In my mind, that's the benefit of a minimum wage program: you might have to pay more for labor, but so does everybody else, theoretically resulting in more customers with cash to spend. As a company which already pays above minimum wage, they happen to get the benefit without the cost.
I don't think they are really the bad guys. Just that it is a lot easier to advocate a jump in the minimum wage when it doesn't effect your payments. They would be the good guys for having these better wages in the first place.
