Is the damp/condensation high up or low down? Is it on an exterior wall, or a party wall shared with your neighbour?
Is there a letter box in the door? Any ventilation through the letterbox? How well does the door fit? Is the top of the stairs open to the rest of the flat?
The problems are that moist air is lighter than dry air, so moisture laden air will tend to drift up the stairs. But cold air is heavier than warm air so will tend to sink to this low point (probably condensing the moisture in the process). So one would hope to have cold dry air around this area. So it might be damp or it might be condensation, hence the questions.
By ventilating this area there's no guarantee whether the cold air will enter there, or the air will exit there, thus pulling the warm moist air from upstairs, which could exacerbate the problem. Normally air would enter at the low point and exit at the high points, but prevailing winds etc do not guarantee this.
Any air entering there from outside will have the same moisture content as the outside anyway, which could also exacerbate the problem.