So, let's say that you're a consumer. You like cheap goods, you dislike sweatshops. How do you encourage the availability of the former while discouraging the latter?
Well, imagine you are a producer of cheap goods, and that those cheap goods are produced by low-skilled workers in an environment that would be considered unpleasant by Western standards. Factory space in your third world country is pretty cheap, as are raw materials. Chances are that a major portion of your expenditure is labor, cheap though that labor may be. Worse, that's not a fixed cost - scaling up your business means scaling up your labor. For every marginal good, you pay a marginal increase in labor.
In the short run, that increase is linear - double the output, and you double the labor costs, hiring more workers, or having workers double their hours. In the long run, the scale is worse than linear - beyond a certain number of workers you need to increase levels of management, consider housing, bigger or more locations, etc.
As an alternative, perhaps you might consider automation. Robots scale much better than humans do, in terms of space and cost. The significant costs with automated production are largely fixed - the initial outlay to purchase the equipment, and the ongoing cost of upkeep. The marginal cost is relatively low - once you've got a well-maintained factory line, the n+1th product doesn't add much cost onto the nth.
A few issues here. You need an area with the infrastructure to support automation (electricity, security, etc). You need to have the money for the initial setup, or some form of loan. You also need to sell a minimum number of units a year in order to support the fixed costs of maintaining your factory and, if necessary, re-paying your loans.
The more units you sell, the more revenue you get, and the worse the bottom line gets using manual labor. Naturally, you'd want to deploy some of your new revenue toward automating. So... our consumer ought to want you to sell MORE goods rather than LESS, so as to push you over the break-even point on an investment in automation, right? Basically, a boycott is moving incentives in the wrong direction, but a purchase is rewarding behavior that you don't want, right?
Well yeah, but what of the workers currently employed? They're working a crappy job because there aren't any other jobs. As the sweatshop sells more units, they're pushed even harder, until automation makes sense financially, at which point... they're out of a job. So they take an even crappier job... Do we imagine that all of the sweatshop workers will become robot repairmen?
Note that unions and pro-labor laws can hasten this break-point in individual cases or throughout a nation. Restrictions on daily and weekly hours, minimum pay requirements, safety regulations and the like all increase the cost of the marginal worker and, by extension, the marginal unit. The higher those marginal costs get, the better automation looks.
In the long term, theory has it that manual laborers can make the transition from factory work to professional careers, working in support of a relatively more automated factory setup. That said, the US is a first world country with a world class university system... and look at Detroit. If we can't manage this kind of a transition, what hope have, e.g., China and India? Is it the manual labors who will make the transition, or their children?
I think that the crux of it is that if we don't want folks working in crappy conditions, it's on us to conjure up a way for folks of their skill, education and location to live a better life when otherwise employed. How to increase the human capital of a population, as quickly as possible? Perhaps we need robot professors.