In terms of contemporary characters that have golden age connections there are still a few around. In terms of the most popular characters though there are not as many. DC really has the advantage here having been around for so much longer, but even then from the golden age the only characters with origins more or less intact are Superman, Batman and Wonder Woman. This is due in large part to the science fiction reboot of certain characters in the 1950s and 1960s (which would be the silver age by DC standards. For Marvel really the only one that remains is Captain America, by grace of having been with a company that was a predecessor to Marvel. In looking at the characters though it is interesting to tie in aspects of their origins which still are tied to the era of the early 1940s
Definitely the connection to Superman is that to Smallville. Coming from a time when comic book cities were given names that either indicated they didn't exist or were descriptive in some way, Smallville by its name conjures up the idea of small town life which was much more common in the time than it is now. At the time the USA was still only 60 years or so away from having "conquered the West" and so Westerns and such minded settings were more popular. Yet even then the towns faced the same problems they do now, mainly that urbanization is slowly destroying them. So Smallville evokes a kind of quiet life that is uncommon anymore in the USA (and increasingly every part of the world)
It might be easy to say here that Batman's link to the golden age is Crime Alley, but in terms of how Gotham is depicted there was not much more crime back then than there is now. Really in the case of Batman the strongest link is to the movie "The Mark of Zorro" which despite the fact that it is a very old movie is still considered the canonical movie that they were watching before things went bad. When Batman Begins showed that the Waynes had been at an opera instead of a movie I thought it was a nice twist, but at the same time that this part of his origin was missed.
Ze Germans! Really in terms of Captain America this is the defining aspect of his origin overall in addition to the super soldier serum. The serum though is not all that different from a lot of other heroes, but the ties to National Socialists is one which refuses to go away despite the fact that this would put most of the characters associated with Steve into their 90s by this point. Of course this is explained by the serums themselves, but as long as there is a Captain America, he will have nazis to battle against.
This becomes tricky. The fact that she is an Amazon does in a way give some connection to the era, but really being an Amazon is no different than being from Mars or Atlantis or Asgard. Until recently I would have said the fact that she is made of clay, but while that is something which is associated with her it is not really in any way related to the era, nor is it even applicable anymore (though non-comic fans who know this about her will still be saying it is the case until at least 40 years from now.) It could be said to her star spangled banner inspired costume, but even that is not the case anymore. I would say then that the connection here is that of pacifism. Not everyone agrees with me that Wonder Woman is a pacifistic hero, but in the early days she was depicted as such even though she occasionally had to back that up with some violence. This has changed in its application over time, it started as a more rigid concept because of Amazon society, but later this does become a defining aspect of her character, sometimes even despite her connection to the Amazons.