Is this something culturally conditioned and left open by the Quran; or is it definitely sanctioned by it?
While this may be more representative of the communities I've frequented rather than the Muslim world at large, I don't know that I have ever seen a "Men-only mosque." While it would not surprise me that such buildings do exist, I think it is a bit of a misnomer to state that "most" masajid are designated thus.
That said, there are a number of wide-spread practices that could lead to the mistaken impression of a male-only masjid. I will summarize them in brief, as in-depth details on each are probably complex enough to warrant separate questions:
- Internal Segregation: Many masajid maintain strong gender segregation within the prayer hall. Depending on the size of the masjid, this may take the form of a distinct prayer hall/balcony for women, or just having the main hall partitioned with a curtain or some form of barrier. Often in such buildings, the entrances are also separated, so even if you see "Men Only" on the front door, chances are there's another "Women Only" door somewhere.
- Cultural Norms: There are many cultures in which women are generally discouraged from unnecessary public exposure. Women from such cultures (or raised according to such cultural norms) would just not be inclined or encouraged to attend the masjid. So even if the masjid itself is open to men and women alike, the fact that no women ever do attend would make it (de facto if not de jure) men-only. (see also this question)
- Limited Space: Some masajid are just plain too small to support the whole community, especially during the major congregations. While the Friday jumu'ah prayers, for example, are generally recommendable for everybody, attendance is only ever considered mandatory for men. As such, if the masjid is too small to hold everybody, there may be a temporary "men-only" policy in place to ensure that these mandatory prayers can be accommodated. However, such policies are typically only during such high-traffic periods, and only insofar as they are necessary.