Ducati was at a real low in terms of both company turnover and product reliability in the run up to 1996.
The American venture capital company, Texas Pacific Group (TPG) then jumped in with a whole load of cash and started putting in lots of quality control to improve the products.
From this point on, Ducati really turned itself around in both the quality and amount of products it could produce, mainly thanks to this cash injection from TPG.
There were a lot of problems with the mechanics on the 95/96 bikes at this low point, mainly due to poor quality components. As this coincided with the launch of the 748, these early models are sometimes best avoided unless they've already been sorted.
All the 916/996/998/748 range have suffered from the same problems which are down to either design faults or bad batches of components - these include the rockers (poor chroming), coolant tanks (these split), oil breathers (these leak), voltage regulator (these weren't "beefy enough"), alternator connector (as volt. reg.), oil pressure switch (poor design), and so on.
Nearly all of these shortcomings have been sorted as time has passed because components get redesigned or manufactured better. As Phillipe says up there ^^^, the 99+ bikes featured most of these revisions so are the safest best for quality and reliability, but obviously at the premium of price.
Don't think of Ducati as any different from a Honda, Kwak or Suzuki to name a few. Most of the Jap models have suffered with their own design faults at some point (Hayabusa subframes, Kawasaki carb icing, etc) but because there is an abundance of these bikes and models you don't hear about it so much.
As with any bike, European or Japanese, check it's history and how it has been serviced. Any competant owner will keep bills or service records for things like replacement parts or work done under warranty. You can then check this and see if you're buying a bike that's about to go wrong or has already had all it's problems fixed.
Obviously, we're all biased to Ducatis but don't forget that as owners we also have first hand experience of it. I expect all of your tales of poor reliability have come from people who have never ridden one, let alone owned one!
There's plenty of great advice on here, so stick around and you might just get to like it...