what do you do with those cities that are completly corrupt (i.e. 1 shield, 1 gold) and are beyond the help of a policeman?

I rush an aquaduct and marketplace in them, hospital if required, make irrigation and railroads all around them and then leave them with lots of happy citizens to earn me lots of points. :) Normally largely undefended as well.

I go Communism, and under Communism, make sure all your Cities are large, as in larger then population 20. This way they make more spt (shields per turn). Once they have more spt, build marketplace to make 'em happy, courthouse and Police Station (I find esp. a PS helps a LOT), then with the excess population, make policemen/Civ.Eng.

I possibly hire a CE to get an aqueduct/market/harbour or so in, but I generally hire scientists/taxmen. Of course, some citizens need to work food tiles (so irrigate the tiles) to keep the population.
Before steam power, I usually order workers there.

I micromanage them to support as many taxmen/scientists as possible. That way I'll get something of value out of the city. If I have Smiths they can have a market place to keep them happy. I usually build a temple to expand the cultural borders. Coastal cities usually get harbours. Other than that you don't want them to grow beyond size 6 as you can't afford the improvements required to keep the people happy.

Ideally they are set to build a worker, timed with city growth. In the ancient/mid they can also build catapults. In late industrial/modern times they can build artillery. When war breaks out just rush them all.

A size 5 city with two railroaded irrigated grasslands can support 3 taxmen/scientists. If there is a food bonus then it be a size 6 city with 4 taxmen/scientists. In large numbers these cities can add a lot to your civ, instead of costing a lot.

Mostly the civil engeer works for me.

So what do you civil engineer users do with corrupt cities if you're not playing conquests? :p

Communism is only useful for moderately-sized empires, or you end up with all cities producing only a couple of shields. I prefer to retain two productive cores around the palace and FP :)

Other -- Edit CivIII.bic/Civ3PTW.bix/Conquests.biq and make the optimal city number 100 (on Standard). Corruption is drastically reduced to realistic levels.

I illigate everything and I fill them with scientists. Will rush a marketplace, aqueduct, hospital as needed.

If they're foreign, I usually raze them. Then again, I'm always under Facism, so I have that "forced resettlement". Of course, I pump out as many workers as needed before razing.

Those tiny, corrupt cities in the middle of nowhere? I use them as airbases/forward military bases since I probably acquired it from one of my neighbors anyway. OR they were by an important resource like oil or rubber. In which case I cease to care about it once I have a harbor and enough defensive units to retain it.

i use policemen. LOTS of policemen and then all my cities have either no corruption or maybe one point.

I chose worker factory.

Culture

I used to throw money at them, and fret about the enemy knights prancing in and out of striking range during peacetime. Lately I've tried to keep an empty swath of (my) territory around a smaller, denser country.

In Vanilla Civ3/PTW, I use them as worker factories. In C3C, I use the new specialists. :)