"Basel" is a international committee of bank regulators who meet in Basel Switzerland now and then. Their mission is to regularize and harmonize banking regulations world wide, with the idea of prohibiting risky lending and speculation of the sort that caused Great Depression 2.0, and preventing countries from taking over international banking thru favorable national regulations. Sort of spread the pain of regulations evenly round the world.
Basel had wanted to enforce a rule requiring banks to have capital (money from investors) equal to 3% of the outstanding loans ("assets" in banker speak). The idea being that capital can be used to cover losses from loans gone bad (lender stops paying on the loan).
The banks screamed and writhed and threatened to hold their breath. And Basel backed down. They changed the rules in complex ways, some kinds of loans don't count, and some derivative deals can be counted as capital, and lo and behold, just about all the banks can meet the 3% standard without raising new capital. Great joy in Euro Bankville.
The Economist article goes on to criticize the concept of a leverage ratio (capital to loans) as crude and inefficient. They prefer a weighted scale where very safe loans ( US T-bills for example) need less capital than say Greek bonds. Which sounds good, but who does the weighting? Reputable US rating agencies like Standard and Poor gave AAA ratings to mortgage backed securities that became worthless.
The Economist likes a more liberal leverage policy. There are two ways to meet a 3% capital to loans ratio. Raise more capital (difficult and expensive) or make fewer loans. The Economist doesn't like option #2, they think it inhibits economic growth.
In real life, at least on this side of the pond, the issue is not all that important. If a big bank gets in trouble, the feds bail it out. We have FDIC, Federal Reserve, and the US Treasury all of whom handed out truck loads of money back in 2007. Bankers like this. In the bad old days, when a bank failed, the depositors lost their savings, and the bankers had to skip town before the lynch mob got its hands on them.