This is not a question Steve would actually answer because doing so would implicate him in ways that (to him, anyway) could be very awkward.
See all the red circles? That class of stuff is “shit that is life-threateningly wrong with Steve.” He’s about 25 pounds underweight and his resting pulse is approaching 90bpm (it should be between 60 and 70), which are things that are life-threateningly wrong even in 2013.
Moving on: rheumatic fever can cause severe cardiac inflammation and cardiopulmonary scarring with damage to the mitral valve causing incomplete flow. This goes hand-in-hand with his general “heart trouble” (possibly either a heart murmur—very likely if he sustained mitral valve damage—or the beginnings of congestive heart failure) and those palpitations I have circled. Palpitations by themselves are nothing dangerous—we all get them from time to time (ever had the shit scared out of you in a haunted house? That pounding in your chest was palpitations). But for them to be listed they’re chronic (recurring), and probably signify arrhythmia. Putting all that into English: His heart has swollen, scar tissue has formed as a result, and he probably has damage to a valve that pushes blood through his heart, so it’s letting blood go backward (that would account for a murmur listed as “heart trouble”) and to make matters worse, his heart doesn’t beat in a normal rhythm, and it’s probably slowly failing. (Having a medical family can be very instructive, kids!) “Nervous trouble” in the 1940s can be one of two things: either he has a disorder in his actual, physical nerves, like sciatica, or he has a psychological disorder (“nervous trouble” = “anxiety disorder”). Since asthma in the 1940s was considered to be a psychological disease and nerves-as-in-the-physical-thing weren’t well understood, I lean toward the latter, which means he had asthma severe enough for them to think he was off his head with mommy issues. (I’m serious. People had some crazy whacked-out theories about asthma.) Guess what the recommended treatments were? Talk therapy and cigarettes. (Not shitting about that, either.)
Why all of this in an ask about suicide? Because all of the above life-threatening shit—combined with those lesser complications like constant colds, high BP, and a predisposition to diabetes and TB—means he could reasonably expect to die by age thirty.
When the movie starts, he is 25 years old.
He’s never been kissed; hasn’t even managed to ask a girl on a date. He has no family who will mourn him when he’s gone—just handsome, well-liked Bucky, who has plenty of girls and probably other friends to quickly make Steve just an old memory.
This is where the headcanon kicks in: Steve would not answer this question because he joined the Army not caring if he died. He wasn’t suicidal, per se, but the distinction is one so fine it would have gone completely unnoticed in 1943, and he would have been considered morally weak and flat-out crazy. But to him, it makes perfect sense: if he dies helping his country, he’s done something worthwhile—and who cares? He’d only have been around another year or two anyway. He would see it as better to lose that year or two in the name of saving people than to lay mouldering away in bed, slowly choking to death as his heart failed. That’s why he was so ready to die when he jumped that grenade, and again at the end of the film—it was no more than he expected when he began.
And so he understands suicidal people a hell of a lot better than he’d like to admit, and he will never say so.