Well, I see you are in Boise. If your friend is not in Idaho, they need to check their local laws.
The Idaho Landlord and Tenant Guidelines put out by the Office of the Attorney General says the following
Landlords must maintain the premises to protect a tenant's safety and health. In that regard, landlords must comply with city and county ordinances and state laws regarding housing conditions.
The following are examples of housing conditions that constitute violations:
Structural deterioration, including cracked and crumbling walls and ceilings and broken or missing doors and windows.
It goes on to mention plumbing, wiring, heating, and so on, but I think the first one is the one that applies.
The landlord is responsible to maintain broken doors and windows. I consider the garage door to be a door. If you had broken it by driving a car into it, it would be your responsibility, just the same as if your child had thrown a ball through a window, but if it breaks through normal use, that is a LL's responsibility, just like if a window latch stopped working.
I will say that I've never had a garage door get stuck open before, they always get stuck closed. When you say it is stuck, are you saying the manual release won't even let him close it, or is the opener just not working? If it is just the opener, is it everything, including the button going into the house, or just the remote? Has he double checked that his sensors have not gotten kicked out of alignment? Has he made sure his vacation setting didn't get turned on by mistake? Obviously, if is just the opener, then the manual release will let the door go down, and the issue with the HOA isn't an issue, but having a secure garage still is.
Honestly though, assuming that the door going into the house has a lock, not all houses have garage doors with openers. Some people manually lift their door up and down each and every time they need in or out. So it could be said that this is not really a security issue, just a convenience issue, and the fact that he rented the house with a working garage door. Of course, if the manual release won't let the door close at all, then it is a bigger issue.
*Edit: Sorry, where I typed "you", replace with "your friend", etc.