Clitarchus' description is given in full on the Livius site. For convenience, I have pasted it into Clitarchus' own page.
Clitarchus' description is said to be an eye-witness account, whether his own or someone else's. The Livius site says baldly that Clitarchus was in Babylon with Alexander, but this is over confident. It is based solely on the Roman historian Pliny the Elder—who was writing 200 years after Clitarchus died.
Contemporary historians immediately assume that Pliny must have had evidence. In fact, he probably had none. Check out the pages on how truth worked and how history was written. Because of the different way that truth and history-writing worked in Pliny's own time, we can make a strong argument that Pliny would have made this claim in order to move Clitarchus' account (and therefore Pliny's own) further up the truth hierarchy.
In other words, Pliny's claim that Clitarchus was in Babylon with Alexander is almost certainly bogus in our terms, although it would have been recognized as a professionally sound strategy in his own time.
So Clitarchus probably never went to Babylon himself, and by extension probably never saw the gardens either. But he had one source about Babylon that is often forgotten: his father, Dinon of Colophon, was also a historian and he had written a History of Persia that is now completely lost. That history may have included a description of the Hanging Gardens.
However, Clitarchus' other sources are well known to have included officers and footsoldiers from Alexander's army. So his description may STILL be based on eye-witness accounts. Now let's assess the quality of those accounts.
Clitarchus' description of the hanging gardens has several significant problems. Where the worst you could say of Herodotus was that he exaggerated, Clitarchus seems plain muddled.
His description is much briefer than Ctesias' description, and also more ambiguous. Clitarchus omits some details and seems to garble others. For example, Ctesias' multiple terraces seem to have become a single one; his supporting vaults have become "stone columns", the hidden water pump is now "irrigation", and Ctesias' citadel, complete with towers, is grafted onto Clitarchus' garden.
This looks like the transmission of a garbled tradition by a non-specialist. As a specialist art historian, I can attest to one very typical mistake transmitted by Clitarchus: most non-specialists are unable to differentiate between piers (that support vaults) and columns (that support beams), and they use the word "columns" for everything. And as an experienced teacher, I can attest to another typical transmission problem: that parts of one description get grafted onto another—usually because the speaker or writer is too hasty to replace the word "it" with something more specific.
So I personally doubt whether Clitarchus saw the gardens himself. I do think it possible, though, that he used eye witness accounts, either from Alexander's army, or possibly from his father, and that the garbling came from his source. I also wonder if the gardens had deteriorated by this time and only the top level was still being maintained.
If you read Clitarchus' description for yourself, you'll see that he does not say who built the gardens.
But later commentators added a couple of footnote-type sentences that got absorbed into his text by even later copyists, so that now people think that Clitarchus attributed the gardens to Nebuchadnezzar. This process is called "contamination", and it is explained in more detail on the page about the builders, which we'll deal with in the next Section.
Despite the fact that Clitarchus is generally considered a more reliable historian than Ctesias and Herodotus, we need to treat his description of the gardens with caution because it is both muddled and contaminated.