I would say men and women are equally irrational. The difference is, when a woman is irrational, you get a frying pan in the face. When a man is irrational, you get nuclear wars, mass murder and billions of dollars of waste. Men hold power, so their irrationalities are magnified and have large repurcussions.
On a deeper level, all humans are irrational. One thing modern philosophers teach is that human behaviour is inherently irrational, and only makes "rational sense" in contexts that are themselves "irrational". Zoom out enough, and all your "rational behaviour" is totally nuts.
A woman's high-heels, long hair and earrings are, for example, absurd and "irrational", but make sense within the context of "acquiring sex", "finding a partner" and "boosting one's own ego". "Finding a partner" is "illogical", but makes sense in terms of the way it placates biological urges, reduces social stigmas (he likes me! We're normal!) and helps you pay your morgage (let's share the rent!). Paying a morgage is irrational, but nevertheless "necessary" given various socio-economic rules. These economic rules are irrational, but obeying them is nevertheless necessary because every other crazy person believes in them. And on and on it goes.
Human behaviour is also largely "meaningless" - what we do has no absolute, Objective Reason behind it. Behaviour only has "rational meaning" within little boxes which themselves are "illogical".
I mean, why the hell are we posting on a West Indies cricket board? Following West Indies cricket is like spending money watching a child abuse victim. Why the hell am I here?
But it's because humans are so weird and irrational that we invent moral and spiritual (even religious) codes. Be nice to people, minimize suffering, share and be compassionate. There is no Objective, Absolute Law which makes these codes "rational"*, but the opposite is often always nihilism, apathy and hate.
* - the Religious Believer will say there is an Absolute Law, which is fine. They simply reach the same conclusion via a different route. Modern science even agrees with them. "Doing unto your neighbour as you would have him do unto you" is a rational act in the sense that everyone ultimately benefits.