It looks like modern cars really are better when you adjust for survival bias
My subjective impression is that modern cars are much more reliable than older cars. But I can see one reason why fans of classic cars might disagree with this impression and it is one of those factors that can seriously bias someone's' perception. It is simply that any old car that is still on the road today is, by a process of natural selection alone, likely to have been on the reliable end of the reliability spectrum when new. The owner of a classic car doesn't see an unbiased sample of all the cars that were produced: she just sees her own car which has survived well (or perhaps the other cars owned by other enthusiasts).
The only way to make a sound judgement is to ask the question in a way that eliminates that survival bias. For example, "how many breakdowns occurred in the first 3 years". With this question you could compare, for example, the original Mini to the current Mini in an unbiased way.
There don't seem to be very obvious places where this history is accessible (but let's see what car-loving skeptics can find: perhaps someone can find the J D Power raw stats on reliability from 10 or 20 years ago). But there is some accessible evidence that points to the conclusion that modern cars are more reliable.
How Stuff Works has a summary which concludes:
...the general consensus seems to be that modern cars don't break down as often as older ones.
The evidence comes from maintenance costs:
The cost of maintenance has also fallen sharply as cars have become more reliable.
Supported by evidence about how long users keep cars before replacing them (which depends partially on their reliability):
A new vehicle-dependability study, from J.D. Power & Associates, agrees, claiming that the average age of a vehicle at trade-in has increased to 73 months in 2009 from 65 months in 2006.
J D Power themselves confirm the importance of this:
Automakers have improved long-term dependability by an average of 10 percent each year since the inception of the study, which is a testament to the industry's commitment to continuously improve and sustain quality, especially long-term quality," David Sargent, vice president of automotive research at J.D. Power and Associates, said in a statement.
I'd like to see longer term data on a consistent metric, but it looks as though the evidence says modern is better.