Men holding doors open for women is still a chivalrous gesture,
even in our less-gallant times.
It should be a natural gesture, so if a woman arrives at the door first and starts to open it, a man shouldn't awkwardly rush in front of her with grand exclamations of "I'll get that!".
Both genders should hold doors open (and check) for people coming through behind them.
It is also courteous for a man to open the car door for a woman, letting her get in first, before walking around the vehicle and getting in himself.