Photo: the front end of a vintage car
The History:
There are, in fact, a number of historical reasons behind men holding the door open for women. Today, we will explore the main reasons: the ancient code of chivalry, and female fashion. In order to get to the heart of this custom it makes sense to begin, at well, the beginning. The medieval code of chivalry demanded that knights/gentlemen protect those weaker, or lesser, than themselves.The custom:

The vulnerable, weaker people would be saved first in this seemingly 'noble' act. For example: on the titanic women and children boarded the life boats first until it was deemed the boats were at capacity
In our modern life of relative safety it is no longer common to find oneself in a situation where harm or danger looms. Women's priority boarding has shifted from the life boats of yesterday, to buildings or cars. Offering to open a door for women is how this tradition evolved.
However, offering to open a car door for a women is a pregnant action. It not only bears all of the historical significance previously discussed, but much more. Men would walk to the passenger door for two reasons: men were the assumed drivers, and women could not always drive cars. These reasons were built upon deeply embedded gendered stereotypes.
The Verdict: LEAVE IT (that door is better left shut)

If someone is right behind you take an extra two seconds and brighten someone's day. But the dated stereotypes and female first mentality should be shut out.
Stay tuned for a new post next week!