How much support does a Core word need?

I don't have a hard rule for this. Some times I have as many as 1 per solar system to as few as 1 per sector. 1 per sector min as the resources from colonies tend to drop to 100% decay if traveling to another sector.

Not all colonies provide the same amount of resources. So a core world might not do much even if you supply it with a dozen colonies.

I try not to get more than a dozen colonies per core world as it'll drive up crime, even if you take basic counters such as the security lights policy.

I favor high class core worlds as they have the tiles needed to build all sorts of districts. I try not to build too many manufacturing districts as they'll drive up pollution. I consider morale districts the most important as you literally multiply manufacturing and research by morale. I try to keep a mix of manufacturing, research, and money after maxing out morale.

I might core a world if they have a rare resource. Its not always necessary as there are techs that allow you to mine or harvest rare resources from a colony world. However, the AI seems to like placing sensors in those slots, and currently there is no way to destroy them without coreing the world (you can fire the governor afterwards).

I echo the above comments from DivineWrath.  If the world is unsupported then manufacturing is going to be tough.  I colonize it anyway.  Then I can then send supply ships or build an orbital manufacturing station to help boost things along.  I usually wind up having that Core World build asteroid miners, Colony ships, Explorers,etc.