I have a Heritage Redfish (14'), and I can stand up on it (haven't stood and cast on it... yet).

So, my experience is that I have no problems with tipping while casting, but this may be a huge function of my boat. But for the most part, most boats that are good for fishing from aren't so tippy you'd have to worry about this (that's just my opinion, I got nothing factual to back it up).

One note... just paddle a little closer to that barely reachable fish. For the most part, I've never worried about spooking the fish while kayaking, until I'm literally right on top of them. I've had some fish within a rod's length from the boat hit the fly.