Captain America is a publicly loved figure, especially among WWII and other veterans. As popular as the Superman, Wonder Woman, and Green Lantern of the day, Cap's niche was to take the fight to the Axis in Europe alongside the Invaders and those members of the JSA and All-Star Squadron who were immune to effects of the Spear of Destiny. Cap openly opposed the Nazi terrorist Red Skull and his forces more than any other 'mystery-man' of the era.
Captain America appeared sporadically after a brief absence in the 50s and 60s, but media access to him was not significant. He eventually faded from public view until the September 11 attacks, after which he reappeared and is once again an outspoken advocate for freedom, liberty and justice. The general populace attributed this to McCarthyism and the Spectre's Judgement, but those with VERY high government clearance might be able to find out there were government-sponsored impostor Captains America post WWII until September 11th and their exposure to the media was purposely limited.
Captain/Commander Steve Rogers is known to some in SHIELD circles as an agent with an Army background. Again, only the very very high up know Rogers is Captain America and that he was found in the ice and revived by SHIELD.
Brooklyn freelance artist Steve Rogers is just a really nice guy, you know?