I will post this as a poll. I will be voting yes, they should. Not only do I think it's common courtesy and a gentlemanly thing to do, but I think I have solid infection control rationale.
Think of it it. If a woman has to put down the seat before doing her business, she will be contaminating her hands, and then she has to wipe herself 'down there'. Maybe she is menstruating and has to perform an invasive procedure with a tampon.
On the other hand, men only have to aim and fire, maybe briefly touching the external part of their organ of urination. There is no chance of the men folk performing an invasive procedure.
I bring this up because it has become an issue in our workplace. Someone keeps leaving the seat up!