I read Big Red by Douglas C. Waller a few years ago. It’s about a post-Cold War cruise aboard the Ohio-class ballistic missile submarine USS Nebraska. Officers and crew with access to nuclear weapons are still carefully screened, and those who indicate even the slightest personal, moral or religious objection to the use of nuclear weapons are steered into other Navy career tracks. The Navy wants to be sure that those who might someday have to use nuclear weapons will be willing to do so When Things Get Very Bad. Repeated drills have the effect, over time, of making missile launches seem almost routine, so that (hopefully) the crew will virtually be on autopilot if an actual launch order ever came in. (For a good fictional example of this, see The Last Ship by William Brinkley, about a Navy destroyer launching nuclear-tipped Tomahawks against the USSR in the late 1980s - a flawed but very interesting postapocalyptic novel).
Waller also said that, since the fall of the Berlin Wall, SSBN captains have discretion to break radio silence and contact HQ for additional confirmation if a launch order comes out of the blue, with no period of international tension or indication of growing hostilities preceding it, and are actually encouraged to do so. Before 1989, captains had no such disretion.