DENVER — Helping the terminally ill end their lives, condemned for decades as immoral, is gaining traction. Banned everywhere but Oregon until 2008, it is now legal in five states. Its advocates, who have learned to shun the term “assisted suicide,” believe that as baby boomers watch frail parents suffer, support for what they call the “aid in dying” movement will grow further.

In January, a district court in New Mexico authorized doctors to provide lethal prescriptions and declared a constitutional right for “a competent, terminally ill patient to choose aid in dying.” Last May, the Vermont Legislature passed a law permitting it, joining Montana, Oregon and Washington. This spring, advocates are strongly promoting “death with dignity” bills in Connecticut and other states.

Public support for assisted dying has grown in the past half-century but depends in part on terminology. In a Gallup Poll conducted in May, for example, 70 percent of respondents agreed that when patients and their families wanted it, doctors should be allowed to “end the patient’s life by some painless means.” In 1948, that share was 37 percent, and it rose steadily for four decades but has remained roughly stable since the mid-1990s.

Yet in the same 2013 poll, only 51 percent supported allowing doctors to help a dying patient “commit suicide.”