I second the question about normal warships helping culture... really? That could be incredibly useful. Does the AI get incredibly pissed if I station an armada in their sector and don't do anything with it?
To avoid a Cultural Defeat: don't get overwhelmed by their culture. Make sure you control at least 10% of the galaxy's sectors, and don't neglect cultural improvements on those planets. Take the culture-improving resources on the map, if only to keep the AI from getting them. Put at least a couple culture-boosting modules on your SBs (I try to have a SB in every sector I own, and when playing for a cultural victory, almost every sector on the map). Deny the AI the cultural wonders like the Galactic Exhibition, Propaganda Machine, etc. Take a cultural bonus as one of your ability picks during game setup.
Doing just some of that should stave off a cultural defeat... doing all of it should result in a cultural victory.
- Ash