Would the act of informing passengers they may die be considered "professional" or "a responsibility of the crew"? No.
And if I may argue, it's the opposite: it is unprofessional for trained flight crew (either pilot or cabin crew) to behave this way.
The shortfalls of this action is obvious. Panic would instantly spread among the cabin. Many people would start behaving irrationally when they believe that death is imminent (hey, not everybody is strong psychologically). This could potentially turn a survivable situation into a deadly one.
In dangerous scenarios, anyone assisting needs to give assertive instructions. We don't tell people why they are doing it. We give very simple, but firm instructions such as "push!", "jump!". Fire fighters and other rescue workers also do it this way.
Telling people "uhhhh, I don't think we can make it" works against the goal of ensuring everyone has a best chance of survival, which is a professional responsibility of any crew member on a passenger flight.
Legally, there are no regulations, industry practices, or airline SOP that I'm aware of, that mandates informing passengers about the potentially fatal outcome of their flight.
Realistically, there aren't that many scenarios where this is possible. If it's a small problem such as high oil temperature or failed hydraulic pump, you won't even notice. If it's a stall or loss of control you'd be screaming already. If it's a inflight breakup you'd loss consciousness in less than 30 seconds. If it's a cabin fire you'd see cabin crew fighting it with fire extinguishers.
That leaves scenarios where the aircraft is stable in the air, but may break up upon landing. Examples would include landing gear malfunctions, flap malfunctions (causing very high landing speed) and total engine failure from high altitude. Historically, most of these incidents end up well. Spending the time to educate passengers emergency procedures (such as counting the number of rows to various exits; briefing how to open the emergency door) is more constructive then saying "On behalf of XXX Airlines, it is with sincere regret to inform you that our aircraft suffered a catastrophic failure and this may be the last moments of your life".
That said, I offer the following:
If you've been told to adopt the brace position, you may consider that it is in the opinion of the flight crew that severe injury and/or death is likely.