If the generator is a marine generator and an OEM or professional installation there is no reason not to run the gene underway. You are rarely heeled over far enough or long enough that there could be an oil problem and both the engine and gene battery chargers should be regulated. If you are worried, just don't run the gene driven battery charger. When running power boats the generators run 24/7 when the engines are on or not; no worries. Even on sportfishing boats, broadside to huge waves while fishing in the Gulfstream, I've never had any problem running a generator.
However, if this is an aftermarket or owner installation, it might be a good idea to monitor things for the first few hours as you have no idea if it is a proper installation.