For the Roman/Greek attitude to depression, we need to look at their ideas about medicine/healthcare.

During a large part of their history, Romans and Greeks practiced a sort of faith-based medicine: illness as seen as a punishment from the gods and as such it should be remedied by prayer and sacrifice.

Traditionally starting from Hippocrates, a more science-based approach arose: the humor theory. The word 'humor' has nothing to do with our modern sense of the word (well, it does, but in a way that goes beyond the scope of this piece), but with the Ancient Greek word for 'fluid', humos. They believed that the human body contained four fluids or humores: blood (haima), sputum (phlegma), yellow bile (xanthe chole) and black bile (melaina chole). The four humours should always be in balance: an excess or lack of one of the humours would lead to illness.

The most important humour for our purposes is the black bile. Having too much black bile would turn you gloomy, uninterested, and listless. Named after the humour that caused this state of mind, it was called melancholia (whence 'melancholy' etc.). The Romans also used the word morosus (whence 'morose').

If we take all references to melancholy and moroseness as references to depression, Latin literature is filled with them. Of course, not all of these are references to cases which we today would classify as depression, but taking into consideration that it's as good as impossible to diagnose these cases accurately and understanding that many of the characteristics of a person a Roman would call morose or melancholic correspond to our diagnostical criteria for depression, it's not unreasonable to suppose that depression was well known affliction, even if they didn't have the same ideas about it as we do.

Whether it was as prevalent then as it was now, is hard to say: I can't think of a way to convert the information in our sources into a somewhat accurate estimate. They're just too biased for that. The sources are written by well-off people about well-off people, and often contain a bias in the content too (e.g., philosophers writing about a remedy for melancholy). Taken with the above mentioned problem that it's impossible to accurately diagnose people based on the texts, there's just to much noise to make an estimate.

Of course, you could ask a physician to venture a guess based on biological and environmental factors, but that's way out of my league as a historian.