Admittedly, there appears to be a connection and Krugman proposes three explanations.
1. CoincidenceHe admits he doesn't have an explanation for option three.
2. Common causation –e.g., neoliberal ideology
3. Actual causation: inequality somehow creates macroeconomic vulnerability
But between common causation and actual causation he misses one: reverse causation. People borrow because they spend it (you don't just borrow for fun, after all; you have to pay interest on that money). With lots of people spending more money, much of it flows into the hands of the wealthy in the form of dividend payments and large bonuses for successful quarterly earnings statements. Yes, some of it will go to workers in the form of higher wages and new hires, but from what I understand about corporate structure, the sum of the lower 99% will be less than the sum of the upper 1% so the very richest get wealthier compared to everyone else.
You could try to tell the story the other way: greater inequality encourages people to borrow so they can "keep up with the Jones's" but people tend to compare themselves with their neighbors, not their bosses. The debt tends to go with the bottom 99% but the graph shows the top 1% with a larger share. We don't really know what's happening to the various income levels of everyone else, again highlighting the use of following an index that includes everyone: not just the very top.