Traveling to Switzerland from the US with a transit in the Netherlands, what is the maximum amount of currency I may carry with me?

There seem to be two different things asked here. The question title is just, "Traveling to Switzerland from the US with a layover in the Netherlands, what is the maximum amount of currency I may carry with me?"

The answer to that is: "As much as you want as long as you make the appropriate declarations in each country."

There isn’t a legal limit on how much currency you can carry, only on how much you can carry without having to tell Customs about it.

Other answers have already generally addressed the limits on what must be declared, but the limits for the U.S. and E.U. are generally 10,000 USD or Euros, respectively.

As Krist’s answer mentioned with regard to Switzerland, so is also the case in the U.S. that, while it’s completely legal to carry more currency than the declaration requirement, it’s a very abnormal thing to do and will raise a lot of red flags. Expect questions about how you got the money and why you’re carrying it in currency vs. transferring it via a bank, which is generally easier and much safer. Make sure you have those explanations and documentation to back them up. Or, better yet, just don’t do this and transfer via banks instead.

Note that, in addition to the rules for entering Switzerland and transiting the EU as discussed in other answers, anybody departing the US with more than USD 10,000 of "currency or other negotiable instruments" is supposed to fill out FinCEN Form 105 before leaving. You can do this online, 72 hours or less before your departure.

(This answer only considers the question at the point of transfer in Amsterdam, it does not consider whether declarations are necessary when departing from the US or upon arrival in Switzerland.)

Your hand luggage may pass customs in Amsterdam. Specifically, this page on the European Commission’s website explains the case (minus the Switzerland complication which I will argue doesn’t change the transfer in Amsterdam):

Are you taking a flight between a non-EU airport and an EU airport with a change of plane in another EU airport? (e.g. Tokyo – Copenhagen – Amsterdam with a change of aircraft in Copenhagen)

You will get off the first plane in Copenhagen where your hand baggage is liable to be checked by customs. Meanwhile, your registered baggage, which will have been given a normal label (no green edges) in Tokyo, will be transferred from the baggage hold of the first plane to that of the second).
On arrival in Amsterdam , your hand baggage will not, in theory, be liable to be checked by customs (Note 2), whereas your registered baggage may be.
If the second EU airport is not equipped for air traffic with third countries (See, for information, the list of international Union airports), your registered baggage will be liable to be checked in the first EU airport.

Your situation is slightly different because Switzerland is not in the EU. Nevertheless, the flight from Amsterdam to Switzerland will depart from the Schengen area at Schiphol. As such, I think the above description will apply because you are going from the non-Schengen to the Schengen area of Schiphol airport.

As such, if you place your money in your registered baggage (which I don’t recommend) then it won’t be liable in Amsterdam. If you carry it in your hand baggage or on your person, then it may be liable in Amsterdam.

As others have said, there are no green and red lanes when transferring at Schiphol, so you won’t have the trouble of going through the green lane when you should have taken the red lane.

Nevertheless, it may be a good idea to have an explanation for the money you are carrying. You probably won’t need it, but it can help if questions are asked.

From my experience with similar itineraries at different airports, you will have to go through a security check. If they see large sums of money (stacks of hundreds for example), I wouldn’t be surprised if that raises questions. At that point, you may have to answer questions because it looks like (even if you’re actually travelling to Switzerland) you’re bringing that money into the Schengen area.

Specifically for the liquid assets, we can look at the page by the Dutch tax agency linked by Mark Johnson. Again, that page does not deal with the Switzerland complication. Using the same reasoning to look at itineraries with a final destination in the European Union (because in Amsterdam you transfer to the Schengen area of the airport), we would expect that declaration in Amsterdam is the normal procedure:

You are flying from Tokyo (Japan) to Paris (France) via Amsterdam (the Netherlands).
You arrive in the European Union in the Netherlands.

If you leave the flight in Amsterdam (the Netherlands) (transfer), you will have to file the declaration in Amsterdam (the Netherlands).
You can use a copy of your declaration to demonstrate in Paris (France) that you have filed a declaration in Amsterdam (the Netherlands) for the liquid assets.

Unless there is some exception for Switzerland, a passenger and their carry-on luggage will enter the European Union (but their checked baggage will not per the first quote in my answer) upon entering the Schengen area of the airport in Amsterdam and leave the European Union (but not the Schengen area) when departing for Switzerland.

Regardless of whether the above answer is correct, it’s a lot of explaining if you get into a discussion with someone at the airport if you decided not to declare it. If I weren’t going to declare because I thought it wasn’t needed but if I wasn’t entirely sure (and I didn’t want to miss my connecting flight), then I would contact Dutch customs prior to my trip to confirm my reasoning. Ideally, I would email them and carry a copy of their response. I haven’t been able to find an email address for them, but this phone number + 31 45 574 30 31 is the Dutch Customs telephone number for private clients calling from abroad.

Switzerland does not require you to declare at the border. But they may check and if you have more than 10000,- CHF on you will ask you where you have to money from, so you should be prepared to prove that you obtained it legally.

https://www.ezv.admin.ch/ezv/en/home/information-individuals/bans–restrictions-and-authorisations/cash–foreign-currencies–securities.html

If you intend to deposit it in a bank be prepared to answer questions from your banker.

The EU requires you to declare sums over 10000,- euro when entering or passing through the EU. But you will not be entering the EU. You will pass through immigration in Amsterdam, but customs only in Zürich. I am not sure that if you stay in the transit area at the airport whether that will count as "passing through the EU" or not.

You have to be aware that travelling with a lot of cash is not something people here do. The normal way to move money from A to B is to do a bank transfer. Travelling with a lot of cash is going to raise quite a lot of red flags. Be prepared for some probing questions when entering the country.

In your place I would look at other means of getting the money to Switzerland. If you wire the money you will still have to prove it is of a legit source, but you will be dealing with a banker who is on your side, not a customs agent.