Amending the US constitution is substantially harder than in most countries: any given amendment must be approved by a 2/3rds supermajority in both houses of Congress, and then must be ratified by 3/4ths of the State legislatures to take effect. The States themselves define requirements for their own "ratification", so they may also impose supermajority requirements (though IIRC none currently do).

Compare this with France, as an example, where constitutional amendment requires either a referendum, or approval both by each house of Parliament separately and then together. Not easy, but you don't need to get the buy in of all the regional governments to get it done. There actually have been amendments in the US which have been stopped by insufficient State buy-in: see the 27th amendment, which was submitted to Congress in 1789 and finally came into effect in 1992 when Alabama ratified it! (Note that most proposed amendments now have ratification-expiration clauses for this reason.)

Amending the US Constitution is obviously possible, because many amendments exist; but the strong cultural significance of the Constitution actually makes it much more difficult to amend. The Constitution is seen as a guarantee of rights and freedoms, and a codification of our "perfect" governmental system. Changing it at all is seen as a strategy of last resort; changing it in any way which is viewed as restricting freedoms, or as trivial, is generally shouted down by members of all sides. The fact that it's a Big Deal makes it harder than the procedure alone.

To balance this, though, there's the fact that it's a lot easier to sideline the Constitution than many like to believe. The only governmental branch with the power to strike down an unconstitutional law is the Court, which cannot strike it down unless someone actually brings a case challenging the law. So an unconstitutional law which is never applied, or which no one hates enough to bring a court case, is likely to stand. And in the US, one must have "standing" with the courts to bring a case--that is, you must have actually been harmed by someone under this law. You can't bring a case if you just disagree on general principles. This makes the process of removing bad laws very slow.

The wording of the Constitution is also fairly vague by intention, and clauses which are interpreted in one way by the courts now may be interpreted differently a century forward or backward in time. This allows a "soft" evolution of its content, and is the basis of much change over the life of the country. Note that a constitutional amendment was once considered necessary to ban alcohol, but that now bans on other substances in legislation are upheld by the courts.

The other side of this, though, is that if a Constitutional decision is made by the Courts, it's seen as pretty much decided by much of the country (though less so recently). Much social change in civil rights, reproductive rights, etc has been accomplished through court challenges.

I don't know how well-decided the constitutionality of the minimum wage is, but it seems unlikely to garner many serious challenges. Note also that the US Constitution includes a clause giving the federal government the power to "regulate Inter-State commerce", which has been used to justify all manner of things.

So to sum up: the US Constitution has a semi-sacred status as the document which enumerates and protects the rights of US citizens. It's pretty damn hard to amend, and this mythic status makes that even harder socially and politically. Enforcement is slow, tricky, and requires court action, but has real results. It's an odd system, I'll admit, and I wonder how well it works sometimes. But I personally like the idea that certain things should be really hard to change, and that those things should include our civil rights.