How can the federal government set a minimum wage? It seems like that would be reserved to the states under the 10th Amendment.
Is the interstate commerce clause? That seems iffy to me.
It's the interstate commerce clause. Because of how the modern economy works. there's really pretty much nothing that's confined to a single state in its effects and doesn't affect other states. Almost any law can be justified as a regulation of interstate commerce these days, especially if that law is actually directly about commercial activity.
The big case to cite here is United States v. Darby (312 US 100). This case tested a federal minimum wage law which banned shipping goods in interstate commerce where they were produced by people paid less than minimum wage, and also banned the employment of those people at less than minimum wage in that context. The Supreme Court found this law constitutional, even though the manufacturing employees weren't personally engaged in interstate commerce, because the employer was engaged in interstate commerce and regulating production was a permissible way to regulate that commerce.
Now, after that decision, there was a major broadening of what was allowed to be regulated under the interstate commerce clause. In Wickard v. Filburn (317 US 111), the Supreme Court ruled that the appropriate test to see whether something could be regulated as interstate commerce was whether it had a "substantial economic effect" on interstate commerce. Under this rule, federal regulation under the commerce clause is very far-reaching. It supports extending a minimum wage to domestic service workers (Marshall v. Rose, a Fourth Circuit case from 1980); it supports extending EEOC rules to state employers (EEOC v. Wyoming); it supports a lot of things.