Semi-submersibles--which is what Sea Launch was (former Ocean Odyssey)--are pretty stable. The marine risers (i.e. the big pipes that connect to the subsea blowout preventer) don't tolerate a lot of angle. Source: Used to work on, among other things, offshore drilling rig mooring analysis.

But I can imagine all sorts of reasons why using an offshore platform introduces a bunch of challenges that launching from land doesn't have. Though of course, the ability to get away from people is a plus too.

One of my wife's relatives was the master for the sea trials for one of the ships involved in Sea Launch. He had an incredible career, starting off as a brass helmet diver for the Royal Navy and ending up with Sea Launch.

That's why SL used most sophisticated rocket of time - Zenit, which was by definition extremely tolerant to platform instability (most other space rockets was not so tolerant).