Why did the Allies let Hitler break the Treaty of Versailles?

The Versailles Treaty was predominantly in France's interest (note the place of its signing). The US and UK initially acquiesced in the wake of World War I, but this acquiescence diminished as time went by. By the 1930s, Germany was no longer at the top of the "threat list" for Britain, at least. France, with Europe's largest army (now that Germany's had been suppressed) and its long Channel coast was in some ways more threatening.

Britain was willing to treat Germany as "another" nation with military parity compared to "others," (not Britain). Specifically, in 1935, Britain negotiated a naval treaty with Germany that allowed the latter a navy 35% the size of Britain's. This actually exceeded the limits allowed to France and Italy by the Washington Naval Conference of 1.67 to 5, or 33% of Britain's. More to the point, this was well in excess of Versailles Treaty limits that allowed Germany only a handful of cruisers, destroyers, and torpedo boats.

I think it is important to understand the environment in the rest of Europe at that time.

Spain had a civil war 1936-1939 (some considering that it was a test for WW2) Italy was under control of fascism.

But lets talk about more "important" countries, in England, the primer minister at the time was more inclined to negotiate rather than to attack, the general idea was that Hitler would be a reasonable person and, worst case scenario, Hindenburg would control him.

Finally France was HIGHLY unstable from a politically point of view with several presidents in few years and a highly polarized society (where people would rather accept a German or a Soviet if that would rule out their enemies).

So it was a context in which European democracies where really weak, and regarding the USA, they were busy enough trying to control the problems generated after the wall street crack in 1929.

To summarize it, the context was really good for what Hitler did, no one would really bother him in his plans, unfortunately.

Concentrating on the Rhineland as a major breach of the treaty, Britain and France had three choices.

1) War. This was out. A lot of blame has been heaped upon the politicians for that, but the populations of these countries, as well as their colonies and allies, were firmly opposed. In Britain, neither the Opposition nor the government nor the public agreed with Churchill's analysis until the destruction of Czechoslovakia.

2) Economic blockade. In modern times countries are relatively good at organising such things. It still takes months or years to organise them, and they're leaky, too. Britain and France had no way of making other countries go along, and would have suffered economically themselves.

Think about Iran: even with every one of the world's most powerful countries and the United Nations on one side, and Iran on the other side, it has still taken almost two decades to get them to back down. Britain and France had nothing like the power over Germany that the USA, Russia, China, Britain, France and Germany have over Iran today.

3) do nothing.

3 is what they did.

The issue with any treaty provision is what will you do if the side does violate it. Ideally, you would instantly spring to war. However, will your allies and your own people support this?

Hitler was able to spin the 100000 man army and the limits on equipment into a straitjacket that wouldn't even let them defend themselves against their smallest neighbors. The Rhineland occupation was touted as a hostile takeover of a good chunk of their land. Many other countries, and parts of the population of England and France recognized enough validity in this that the leadership felt it wasn't worth going to war over.

The wording of the question betrays the bias of hindsight. The idea that Hitler could have been brought to heel by decisive collective action in the mid-1930s has tremendous appeal now. But at the time rigidly upholding the terms of an unworkable 20-year-old treaty would have seemed to most people to invite disaster not avert it.

Breaching the treaty

It isn't as if the treaty's terms hadn't already been breached, fudged and watered down even before Hitler's rise to power.

Hitler wasn't the first European leader to thumb his nose at the treaty in general and the French in particular. In 1923 little Lithuania engineered an occupation of the Memelland and effectively ousted the French authorities there (French administration of the territory had been mandated by the Versailles treaty). The Lithuanian action was accepted as a fait accompli by the international community.

An early exercise in strictly enforcing the Versailles terms was the French/Belgian punitive occupation of the Ruhr in 1923. Its objective was to force Germany to keep up reparation payments. It wasn't a success. The French action was seen as heavy-handed, and wasn't repeated.

In fact the schedule of reparation payments mandated by the treaty was never kept to. Pre-Hitlerite Germany had already revised down the reparations payments agreed to at Versailles, in 1921 and 1924. Arguably Germany was thereby defying the terms of the treaty long before Hitler's rise to power.

The same democratic German government was also breaching the limits on the size and scope of its armed forces, with the British and French turning a blind eye.

Unreasonable Versailles/Reasonable Germany

It's important to remember that the Versailles Treaty was a harsh peace, and was perceived as such. Not only in the defeated Germany but also, gradually but increasingly, amongst the victors. Keynes in 1920 called it a "Carthaginian Peace". According to Keith Robbins in History, Religion and Identity in Modern Britain a "certain shame" emerged at how even scholars had talked about the "Huns" twenty years before.

If you perceive the Versailles treaty to be unreasonable it's a very small step to perceiving the German demands as reasonable, even sensible.

Prior to Hitler, Germany had a decade long record of good behaviour, at least on paper. The Locarno Treaty and other treaties had effected Germany's diplomatic rehabilitation. Within the European democracies distaste for Hitler wasn't incompatible with the general feeling that German grievances were far from groundless.

The "Allies". Which Allies Exactly?

Ah, so it was job of the "allies" to force Hitler to back down. Which ones, exactly? Victory in the Great War had been a collective effort. Roughly speaking, the victorious allies responsible for the Versailles treaty included Japan, Russia, Italy, the United States, Britain and France.

So an operation to bring Hitler to heel and enforce the Versailles treaty? Japan and Italy? Nope. The USSR and the United States? No, and explaining why not would require a whole new set of questions and answers. So suffice it to say that we are talking about Britain and France.

Viewed from Britain, and more especially France, the job of upholding the Versailles treaty was beginning, by the 1930s, to look like something that everyone else wanted to volunteer France for.

Climate in Britain & France

Which brings us to the political and social climate in France. Not only was there instability at the political level but there was also a range of other problems. More than any other country in Europe France had been left exhausted by World War One. The currency had been left weak. The declining birth rate was a source of constant concern, so much so that prime minister Briand stated "Our birth rate dictates the foreign policy I make". These were France's so-called "hollow years". In the face of a resurgent Germany French solutions included accommodation/appeas*m*nt, turning away from Europe and towards empire, retreating behind the Maginot line. Confronting Germany depended heavily on a network of alliances, including at various times Britain, Russia and the smaller European nations. But these alliances were all problematic and burdened by mutual suspicion.

The British had as little appetite for confronting Germany as the French. The famous "King and Country debate" is often cited as an example of the pacifist mood in establishment circles in Britain. Equally famous and relevant to your question is Neville Chamberlain's objection to involving Britain in a "quarrel in a far away country between people of whom we know nothing". Britain's Ten Year Rule restricting rearmament indicates Britain was very far from being ready and able to counter technical breaches of the Versailles treaty.