Here’s a predicament that touches most of us at one time or another: What happens when you’re too sick to go to work? Or a family member is so sick that they need your care?

Among industrialized nations, only the U.S., Canada and Japan do not mandate universal access to paid sick leave. Some employers in these countries voluntarily offer it as a fringe benefit. In the U.S., according to economists’ studies, about 65 percent of employers provide it voluntarily to full-time workers. But less than 20 percent of low-income, part-time and service sector workers are covered.

Eleven U.S. s...