I get asked this question rather often. Here is my answer, and my reasons for it.
The underlying assumptions behind the question (which like many assumptions are gross over-simplifications and largely wrong) are that women are physically weaker than men, and less aggressive. Either one of those is apparently a disadvantage in a fight, so might cap your performance at a level so low it makes participation a waste of time.
Let me be absolutely clear: the WHOLE POINT of martial arts is that skill beats muscle. Only when skill is equal (or you are unskilled) does brawn make a difference.
The WHOLE POINT of swords is that they are labour saving devices. It takes very little strength or power to kill someone with a sword.
And the WHOLE POINT of training is that IT WORKS. The weak get stronger, the timid become more bold. The rash learn caution, and those that have relied on their strength learn other ways to win against the day when their strength will fail them.
Let's leave aside the simple facts that a third of my students over the last 15 years have been women, that many of the class leaders and instructors I have trained are women, that the only instructor I've ever invited to teach medieval wrestling at my school is a woman, and that I can think of at least a few women who could soundly kick my arse in wrestling, or unarmed combat, or with a range of weapons from dagger to sword to bow to gun. These women by themselves prove beyond reasonable doubt that women can be excellent martial artists and swordfighters.
However, the question is not, is never, “can I be as good as that person?”. It is “can I be better than the person I am today?” Yes, obviously. I've never known the answer to this question to be “no”.
In asking that question, we can then ask “will swordsmanship training make me a better swordfighter?” The answer to that is invariably “yes”, assuming a decent teacher or group or school.
And the question after that is “will becoming a better swordfighter make me a better person?” That can only be answered by the whisperings of your own heart. If I didn't believe that the answer is often “yes”, I wouldn't teach swordsmanship for a living.
I extend the exact same logic to anyone and everyone, regardless of size, age, (dis)ability, or any other thing. And it makes me furious beyond reason to think how the assumptions of the question, and the frequency with which it's asked, imply that women are so generally assumed to be ‘weak', ‘incapable', or in some critical way inferior.
I've taught a few people to shoot pistols, including my sister and my best friend. My best friend is an experienced martial artist, stuntman, bodybuilder and all that. My sister is a copywriter, with no weapons training or combat training of any kind. My best friend is the only person I've ever seen who actually managed to hit both the floor and the ceiling of the range in the same session. My sister got every shot onto the paper at 25 metres, first with a .22, then a 9mm, and then we had some fun with bigger calibres.
The difference? My friend has seen just about every action movie ever made, and couldn't help acting the shooting. My sister just did everything exactly how I told her to do it, as best she could. Guess which one I'd rather have show up to a sword class?
You might also find these posts on these related topics interesting:
Women's Class (regarding gender-segregated classes)
Swords do not discriminate. Neither should swordsmen. (regarding trans swordspeople)
Gay marriage in Finland? About bloody time. (regarding, you guessed it, gay marriage in Finland)