Not all basic labour costs necessarily increase. It really depends on how dire the situation in the country is, which industries are affected how much, how great of an increase it is, and what other parameters the bill might bring with it.

Developed nations are probably those that have the hardest time competing with other countries wage-wise, so let's look at them: Most of their wages are already fairly high, even relatively speaking, which means that usually only a small portion is targeted by minimum wage proposals. Also, there is a great array of imported products, as well as products that are manufactured locally with some imported components.

The industries that are often underpaid are of the service kind, such as restaurant and hotel personnel, cashiers, cleaners or janitors, which produce something that can't really be imported all that well. There are also cases in other industries, of course, whereas agriculture might be one of the more difficult ones in that regard. Manufacturing, however, covers a wide array of jobs and wages, which means that a minimum wage increase would only affect a portion of it, depending on how dire the situation has grown in that sector for the country in question.

Understanding that labour costs are only a portion of the product, and that only a portion of most products's labour cost is tied to labour paid in wages below the (proposed) minimum wage in the country of sale, and the fact that more money in the pockets of labourers affected by the increase means more purchasing power in the populace, the effects are quite less simplistically negative than you're letting on.

At the end of the day, consider this: Earning $4000 a month is considered low in Switzerland, and anything below is viewed as outright abysmal. Sure, costs are also much higher, but at the end of the day, people have good purchasing power on average, and wages in the 5k-7k range aren't uncommon at all, and they're by far not the end of the rope.

If high wages and losing out to imports were such terrible problems, Switzerland would be an economic wasteland by now. Yet, you'll have a very hard time finding anyone who'd claim that the country is doing anything less than "fine" in the economic grand scheme of things.

You might want to have a look at this article, it has quite a bit to say about the minimum wage.
Here's an article having a bit of a look at a breakdown of hamburger costs in a minimum wage context. I suggest having a look for illustrative purposes.