Does the client just randomly shut off? Like your computer is totally fine, but the client just closes itself for no reason at all?
I used to have that same problem all the time on my old computer and I don't know why. The client was a much better gaming experience for me than playing in the browser, but I had to go back to the browser because the client would just randomly close on me. Sometimes after 5 minutes, other times after 2 hours. You just never know when or how many times it will close on you.
My suggestion is to download Swiftkit. Works like a charm, and is a 100x better than the RS client anyways as it has TONS of features that are very useful for when your playing. I would also check to make sure you have the latest version of Java, and maybe clean your computer up a bit so that its not lagging you down or causing your game to crash.
It's probably your graphic card. I had a shite won not too long ago and got black screens every two steps, so I decided to buy a new one. Now it works like a charm.