Do that. When people ask here about dampness, the three questions are always "Do you have a kitchen extractor? Do you have a bathroom extractor? How do you dry clothes?". Fix those and if there is still a problem then investigate more.
I don't think you've proved that. Do you see evidence of penetrating damp coming through the walls? More likely, being a basement means that the walls are cold, resulting in condensation, and possibly that the outside air (which is exchanged via the trickle vents and other "leaks") is relatively cold and moist.