If you plan to work past 65 and keep the health insurance you’ve had from your job, you’re likely to wonder what, if anything, you need to do about enrolling in Medicare.

About one in six older Americans now remains in the workforce beyond what was once the traditional retirement age. And the number of older workers will only grow over time.

One reason is that Social Security now requires you to be at least 66 to collect your full retirement benefits. Retiring earlier means a smaller Social Security check.

Then, too, a number of 60-something workers continue to pursue their careers because the...