It's not. It's not rude to do a nice thing for someone
because you respect or like them. For example, I open doors for elderly people
- of either gender, hold doors open for people that are passing through them at the same time as I am out of common courtesy
, and open doors or give up my seat for friends.
Unfortunately, it is the case more often than not that when I hold the door open for a man, he will grab the handle out of my hand and stand there looking foolish until I precede him through it. That is the height of rudeness - to reject a polite gesture so flippantly because of some sort of archaic idea of chivalry. It's the kind of chivalry that isn't.
On a more general note, a woman at the bank or the entrance to the canteen doesn't know if a man holding the door open for her is being nice to her as a person, or "correct" (according to his own ideas of correctness) to her as a woman. And yes, there is a problem with the second case - because doing nice things for someone chiefly because of their gender is just the flip side of doing nasty things to them for the same reason. And that is a very raw area for many women - particularly in the workplace - today.
My suggestion would be that the safest thing to do (and the nicest!) would be not to run forward and swing the door open in her face, but rather to pass through it and discreetly hold it open behind you for a second or two, so that she gets all the effect of having the door held open for her without the unpleasant side effects of pompous "manners".