What are the arguments for and against multinationals relying on sweatshops in their global supply chains?

Question

Accepted Answer

For: they are often the best locally available job and a rung on the development ladder. Against: the "choice" is made under duress, and the standards are set by whoever has the least power.
The term sweatshop was coined in England around 1850 for production sites where workers labour for low wages under bad conditions, above all in the clothing industry.
Arguments for inclusion:

Wages, poor by rich-country standards, are frequently better than the local alternatives — subsistence farming, informal work, or nothing.
Export manufacturing has historically been a development path: several economies climbed from low-wage assembly to higher-value production within a generation.
Withdrawal is not neutral. A boycott that closes the factory removes the income without supplying a replacement, and the documented result is often worse work elsewhere.
Consumers in importing countries benefit from lower prices, which matters most to the poorest of them.

Arguments against:

Consent under duress is not consent. That someone accepts an offer because the alternative is hunger says nothing about whether the offer was fair.
"Better than nothing" is the wrong benchmark. The relevant comparison is not the local alternative but what the buyer could afford to pay: labour is typically a very small share of the retail price, so a large relative improvement costs little.
Some harms are not compensable at any wage — unsafe buildings, blocked fire exits, toxic exposure, child labour, suppression of organising.
The race to the bottom: if standards are a cost, competition pushes them downwards everywhere, which is precisely the moral-dumping fear.
Responsibility does not stop at the contract boundary: the buyer sets the prices and lead times that make the conditions necessary, which is what supply-chain due-diligence laws are built on.

Notice that the two sides largely appeal to different moral theories — the case for is consequentialist (net welfare, the counterfactual) and the case against is deontological (dignity, rights that don't yield to aggregate gains). That is why the argument so rarely converges, and why naming the theories is the first useful move.
Go deeper:

Wikipedia: Sweatshop — the history of the term and both sides of the economics, with the development-path evidence.
 Ethics Unwrapped: Causing Harm — why harm at a distance, through a contract, feels so much lighter than harm you can see.