Just because women are perfectly capable of surviving without men doesn't mean we don't appreciate some of that old-fashioned chivalry from time to time. As long as you do it with a smile on your face and no expectation of any returned favors, there's nothing wrong with helping a lady out when you want to. Really, there's nothing wrong with helping anyone out when you can, even a fellow dude!
Feel free to open doors, tell us we look beautiful, give up your seat on the subway, and offer a helping hand whenever you can. If a woman refuses, don't take it personally and she'll appreciate you even more. While we're at it, some old-fashioned clothing wouldn't go astray either. We're talking nice belts, crisp shirts, bow ties, and anything else that is totally gentlemanly.