Quoting Paul Krugman is not helpful, because he never learned to speak English! :-) His blog is a vast accumulation of jargon, essentially designed for a University-level course in Economics.
If you want to know what he is really saying, I will attempt a translation -- many of the concepts are valid, some are even cutting-edge. The translation is a bit rough-and-ready, it's a long time since I did an Economics course.
The basic concept is that a monetary policy which works for one nation, Germany say, will not work for 26 nations. This is common sense, even to economists -- but not to the EU elites.
Monetary policy includes such technical aspects as money supply: if you simply print more money, you decrease its value, because you have increased the supply - the amount of money in people's pockets - but not the amount of goods available. So you have more money chasing the same amount of goods, hence you get inflation -- a general rise in prices of all goods, on the basis of supply and demand: demand for goods has increased because people have more money to spend, but there are not more goods to buy. It has become a seller's market, due to a relative scarcity of goods, so sellers can raise their prices.
Money supply might change: this change is good for one country, say Germany, which manufactures goods, but bad for everyone else, who don't. If you make a change which benefits Germany, the richest of the 26 nations, that change inevitably hurts everyone else. And vice versa.
Having a single European Bank, which sets that supply, means that none of the 26 nations has control over that key aspect of their economic wellbeing. Inflation is now outside their control.
Interest rates are another aspect of monetary policy: if you raise or lower interest rates, you make your currency more attractive, or less attractive, to foreigners - so the exchange rate fluctuates. If it goes up, your exported goods become more expensive, which harms your exporters; or if it goes down, imported goods become more expensive, which harms your importers.
Normal countries choose an interest rate to balance the competing needs of those domestic businesses which are nett importers, and those which are nett exporters. But individual nations lose the power to do this if you have 26 nations but only one currency: with a single interest rate, applying to all 26 countries, the chosen rate never will be good for them all, because some nations such as Germany are nett exporters, and others such as Greece are nett importers.
Krugman's other point is that there is no mechanism for adjustment.
In broad terms, what this means is that having 26 nations but only one currency takes away the former flexibility.
In the old days, each country could set its own exchange rate applicable to its currency, as each had its own currency. That gave the 26 nations great flexibility. It provided - in the jargon - a mechanism for adjustment. In other words, each nation had an economic mechanism for adjusting its own exchange rate.
The importance of this is that in bad times you can depreciate: by various means (complicated, but very effective) a nation could reduce the exchange rate of its currency, so that from the point of view of foreigners its exports were cheaper, so it could sell more goods abroad (as they are now more competitive in overseas markets) and so make more money. There was a cost: imported goods were now more expensive. But this, too, was beneficial: domestic consumers would buy more domestic goods if the price of imported goods rose, which would benefit the domestic economy by increasing sales to domestic customers also.
The benefits of depreciation are enormous, to an ailing economy. It can mean maintaining real domestic prosperity, and avoid a damaging slump or even an economic collapse.
This enormous benefit is lost when you abandon your own currency. As a single nation sharing one currency with 25 others, you fight - and lose - if you want to devalue, because although devaluation helps poor countries, as described above, it hurts rich countries such as Germany, as they don't benefit from it.
Rich nations want to move the exchange rate the opposite way: to increase their wealth by increasing the exchange rate instead of reducing it. Germany imports raw materials for manufacturing, so it is harmed if the price of imports increases through devaluation.
The other really bad news that comes with having a currency whose exchange rate you don't control is that, within the bloc of 26 nations, you can never adjust the exchange rate between you and, say, Germany. Whatever rate each country joined at, on initial conversion to the Euro in 1997, it is stuck with forever.
All 25 are competing with Germany, the 26th member, which is the most powerful economy in the bloc. But now they cannot keep up with Germany by the least painful method, their former ability to devalue their currency. They now share a currency with Germany, and have found that the effect is as if they had abolished their own currency and adopted the Deutchmark!
German productivity, German wage-rates, German interest rates: the other 25 nations have to meet Germany on a level playing field, in respect of productivity, wage-rates, and interest rates -- even though their own economies are much smaller than Germany's, and so much weaker. And all because they have lost their principal means of competing with German industry: the ability to devalue their own currencies.
In a normal country, the central government sets tax rates for the whole economy, raises its public revenues from those taxes, and then spends the money -- often on helping areas of the country which have particular economic problems.
This is a means for raising money from the richer parts of the country, where more people have jobs, and where more people have better paying jobs, so pay more taxes; and spending that money in so-called 'depressed' areas, where fewer people have jobs.
There are various means of doing this: paying unemployment or welfare benefits is the main one. It's a transfer of money raised in economically prosperous areas, through taxes, to areas where there are fewer jobs and many people are unemployed.
This doesn't happen in the EU. There is no central government, no mechanism for raising money centrally: income tax and capital taxes are raised locally in the 26 nations, and kept by the 26 governments. So there is no welfare, and no dole payments, except within each nation.
This means that poor areas of the EU stay poor, because money is not being constantly redistributed by an adjustment mechanism, from richer areas such as Germany.
When Greece got so poor that it was staring bankruptcy in the face, it had to receive a massive bailout, and a huge write-off of its bad debts: this is the kind of huge economic shock that can hit a small nation of the EU, because the EU tries to behave like a normal nation state but lacks the financial arrangements which would permit that.
Small, poor nations are in full economic competition with Germany, on a completely unprotected basis, hence are economically damaged by being unable to compete with Germany -- a small economy simply can't produce goods on the scale of the massively powerful German economy, so can't compete with Germany on price, so will always end up forced to sell its goods (probably manufactured at a higher cost than in Germany) at a lower price than German goods.
So Germany gets richer, and everyone else gets poorer. Until eventually the minnows - the smallest and poorest of the other 25 nations - will need a bailout. And then the Germans will traditionally complain at being asked to pay for the bailout!
The real answer is that if the EU was a nation state like the USA, with a central government in control of taxation, welfare, interest rates, monetary policy, and exchange rates, its currency would be strong like the dollar.
The USA became a nation state through having a common language, a common origin, a common religion, a single economic system; and because the states shared a common history and culture, they generally pulled together.
The EU has none of this.