I always made my routers remotely accessible when I could. It was very important for me to test things.
From what I've heard, there are now well-known security exploits affecting most of the popular older consumer-grade routers. The exploits I've heard of don't require remote administration to be enabled.
Anyway, I have no way to access my current router (Apple AirPort Extreme 5th gen) remotely. From what I can tell, you need a Mac to do it. I can't even do remote access from the iOS app. I just connect to my PC using Remote Desktop Connection and use the AirPort utility from there.
Damn Apple won't give it a web setup...grumble grumble.