As most have said, I always hold the door open for anyone. I also say 'don't mention it' if I get no thanks.
Leading on from the OP and the risk of being accused of being sexist, agest etc, I always find this problem with giving up my seat on buses and the tube. There are a lot of people who get offended if you offer your seat to them. I'm not sure if there's meant to be some sort of age restriction where you only offer your seat to people over 65, but short of asking their date of birth it's hard to tell!