The nature of the evidence is such that the question is more often the other way round: 'how can we be sure this text is wrong?'

What I mean by that is that, very often, the burden of proof lies with someone wanting to doubt the accuracy of a text. There doesn't need to be much proof. But the presumption is that a copy of a lost original is, well, a copy.

Like I said you don't need a strong reason to doubt its accuracy. But you do need to have some kind of reason, whether it's because

1. there are variant readings in different manuscripts of the same text
2. the text as it stands doesn't make sense
3. information in the text conflicts with other known data
4. another text appears to be familiar with your text, but they seem to have known a different version of it
5. spelling anomalies
6. stylistic mismatch with context
7. problems with the rhythm of the text
8. you can think of an alternate that would work better, and there's a plausible scenario where the text might have been altered
9. a reading could be explained by a known type of scribal error
10. there's reason to suppose scribal interference

or something else that makes sense. Like I said, you don't need strong evidence, or a strong argument, to infer textual corruption. But you do need some kind of evidence or argument.

As it happens, in situations where it is possible to compare the mediaeval manuscript tradition with an ancient copy -- generally a very, very fragmentary ancient copy -- the errors are typically very minimal, and tend to be either (1) the same kinds of errors we see when a text survives in multiple copies transmitted by the mediaeval tradition, or (2) differences that are a direct result of practices that were normal at the time the ancient copy was written, especially spelling variations from before Greek spelling got standardised in the Byzantine era.

I wrote a longer version of this, with detailed examples, offsite a few months back. I'm explaining this here rather than simply referring you to the offsite piece because I want to adjust the emphasis here.

Everything I've said here is true for most texts produced in the late Hellenistic and Roman eras. But there are some texts, especially texts that were originally written before the Hellenistic period, where there are additional problems. In these cases there often is more reason to be sceptical of a mediaeval copy. But even here, that's because there is a reason to be sceptical.

For example:

1. Classical Athenian drama poses special problems because actors adapted plays in performance, improvising lines, swapping lines round, or borrowing lines or whole passages from other plays. This kind of interference can sometimes be detected in the text, and there are bound to be many more cases that we can't detect.

2. Editions of classic texts in the Hellenistic period, like Athenian drama and Homer and Pindar, could sometimes be the result of scholarly editors of the time making an effort to standardise those texts and iron out wrinkles that may have been present in their original form. We can't generally spot alterations of this kind.

3. Much pre-400 BCE literature had to be rewritten using Ionic orthography after Athens adopted the Ionic alphabet, and we have no direct line of sight on how this process worked, who did it, or how much the texts may have been altered as a result.

4. Homer is a special case for lots of reasons. All three of the above points apply to Homer, and there are more problems too.

But I think the important part of the answer is that, like I said, where it's possible to compare an ancient copy with the mediaeval tradition, the mediaeval tradition is accurate enough that we don't start out by thinking of it as full of corruptions: rather, there needs to be some kind of evidence or reasoning to suggest that we're looking at corruptions. That's just where the burden of proof lies. It can be a really weak argument, but it does have to be an argument.