This doesn't make any sense (though you have paraphrased Sanders accurately). While technically true for some definition of "conduit of welfare", the same is true of almost everybody in society. Welfare recipients are integrated into the economy enough that "if X didn't get welfare, person Y wouldn't be as rich" is true for many, many, many values of Y (neighborhood stores in poor areas, people benefiting from reduced crime ("stealing bread to feed your family" used to be more than a trope), etc). To give an example in microcosm: if you buy a used car on Craigslist from someone, the fact that the seller is a welfare recipient doesn't mean that you're a beneficiary of welfare because you're not paying him enough to live off of.
To the extent that society is responsible for helping the poor (FTR, I personally believe in this responsibility), it makes absolutely no sense to claim that employers (as opposed to the welfare system) are responsible for filling the gap between "market value of a person's labor" and "how much income he needs to reasonably survive".
This particular claim of Sanders is absolutely idiotic: people are blindly pattern-matching it to support for the poor when in reality he's arguing for shifting the burden of subsidies from all of society to arbitrary consumers/business owners/employees affected by artificial wage floors.