A couple of ways. For nylon strings (syn gut, multifilament), when the strings start to move around a lot and you have to physically put them back into place, then it's time.
Play time. I can get about 15 hours with syn gut. 20 or so with a good multifilament. Less than 10 with poly.
Multifilament strings will have an outer wrapper that will start to fray when they are starting to die, so look for fraying.
Poly usually dies long before it breaks. As it dies, it gets even more stiff. Perhaps your arm can tell you when its close.