Can a requirement in bylaws stating garage doors must be kept closed at all times be enforced?

It really depends on where you live I think. They tried to pass this where I live, but there are many objections so it's still in debate. My cousin in Miami was being threatened with a lawsuit since he wants to keep it open during the day, but nothing really happened so far.

I guess the best advice would be to consult a lawyer and see if there is a way to leave it open with no consequences. That being said, leaving the door open is not really recommended anyway - it's a security and safety risk, and you don't want that, trust me.