Minimum wages will increase in 19 states at the start of the year, a shift that will lift pay for millions of individuals and shed light on a long-running debate about whether mandated pay increases at the bottom do more harm or good for workers.

In Massachusetts, the minimum wage will rise $1, to $11 an hour, affecting about 291,000 workers. In California, the minimum goes up 50 cents, to $10.50 an hour, boosting pay for 1.7 million people.

Wages...