Minimum wage laws (like maximum hours laws and trade union/other pay scales and compulsory membership of trade organisations) are socio-political ideas not economic ones. The textbook economic argument was always that these things shift the labour supply curve left (or up) and therefore reduce the quantity / raise the price of cleared employment.
My understanding is that economists who have looked for this effect have--almost everywhere--found that it is either too small to see or too difficult to isolate (and some studies have no doubt come up with the opposite result). So there is no economic consensus that minimum wages do actually reduce employment in countries that have them (which is just about all the "rich" ones--poor countries like India have statutory minimum wages too but informal employment is so large in the country that it would be even harder to extract any result).
It is possible that the inefficiency (dead weight loss) that is created by distorting upwards the price of the cheapest (legal) labour is absorbed by lower wage income higher up the scale, rather than by reduced employment. That would be the outcome desired by egalitarians anyway, and would be protested by laisser-faire types. It's also possible that instead of reducing employment it raises retail prices so that there is an income loss shared over more of the population. Again, not so bad for those in favour of redistribution. And because both types of redistribution completely avoid tax-welfare transfers, they are a much easier political sell, though this is pretty much because the redistribution is fairly well hidden.
Minimum wages don't help reduce poverty in the case of poor households where nobody has a job, and they would increase it if they raise retail prices. And they're unnecessary (from a distributional equity standpoint) for households that are not poor because a low-earner lives with a high earner. For both of these situations, working tax credit (earned income tax credit / negative starting rate of income tax) is more effective. But it suckks up tax revenue so it's harder to do.
My view is that social welfare transfers are better than market regulation.
