Hang on, ἀργία is not about income, it's about steady activity. The assumption, legally and otherwise, is that such work necessarily yields profit, but there's nothing in what we know of the ἀργίας γραφή, the "crime of 'idleness,'" that says that the accused has to provide proof of actual income. Indeed, our information about this law at Athens (and similar ἀργίας νομοί in other cities) is extremely thin and very repetitive. Mostly the ἀργίας γραφή is brought up to discuss the distinctions between the Draconian and Solonic laws. Both Lysias fragments that mention the law are in this vein, as are Plutarch Sol. 17.1 and Julius Pollux. All these authors are interested in differentiating between the punishments associated with the ἀργίας γραφή, and this appears to have been the important point, whether the condemned suffered ἀτιμία, effectively the loss of citizen's rights, or something more severe, and after how many convictions he suffered the highest penalty. No mention is given of how exactly the case was to proceed in court, except that Lysias says that Solon transferred the indictment from a private one to a public one. Most of the scholarship on the ἀργίας γραφή is interested in whether this is a genuinely archaic crime, or whether it's a later insertion into the Solonian corpus, especially since the crime is not included in the list of crimes at Athens in the Ath. Pol. Additionally, we don't really know what exactly the law was supposed to do, whenever exactly it was included in the corpus. The general consensus is that it's some sort of law protecting household property rights, that it was connected to inheritance law, and that the idea is to prevent agricultural holdings from suffering from an heir who doesn't actually work them.

Indeed, Lucia Cecchet has pointed out that there doesn't really seem to be any reason to suppose that ἀργία even really refers to "work" generally, even if linguistically the word is derived from ἔργον, "work." The meaning seems typically to refer to the lack of agricultural labor, not general activity, and as early as Homer the ἀεργός (=Class. Att. ἀργός, an idle person but typically in Attic also unworked land). Cecchet mentions a couple of passages that may or may not be associated with the ἀργίας γραφή. Herodotus, for example, tells a very unbelievable story about Solon bringing a law from Egypt that required every Athenian citizen to declare publicly how he made his living, and if he refused or could be shown to make his living illegally he would be put to death. I'm somewhat unconvinced that this law existed (certainly the punishment is a fantasy), but the burden of proof here is clear. The citizen does not have to prove anything, but is only punished if his word can be disproven. A very problematic reference in Theophrastus mentions that Pisistratus encouraged the Athenians to work their fields rather than assemble in the city, so that he could avoid revolts and other disturbances. Now, critically, a speech of Demosthenes mentions in passing a law on ἀργία. In the Against Eubulides, the speaker, who is accused of being the son of a non-citizen woman, makes two references to the law to defend himself. First, he attacks his accuser Eubulides, whom he claims makes his living by making malicious lawsuits, the idea presumably being that powerful people are paying him to do this (the similarity with not only parasites but also informers is clear). Second, he points out that the prosecution's case that his mother was a non-citizen rests on the fact that she was a wet-nurse and a market merchant, activities which are associated with metic women in the Athenian mind but which, the speaker reminds the jury, are perfectly legal occupations for Athenian citizen women. Indeed, he mentions that during the Social War many citizen women were forced by necessity to take jobs that would normally have been done by slaves.

That last comment, that these women "needed" to take such jobs, is a little obscure. Does it mean that "unemployment" was illegal, and that in order to show their employment citizen women were forced to take terrible jobs? Surely not: Athenian women did not, as a rule, work, or even collect their own income, and were not expected to. The necessity here must be an economic rather than legal one. But in Demosthenes the fairly strong connection elsewhere with agricultural labor and ἀργία has apparently disappeared. The speaker's suggestion is that Eubulides, in comparison with the speaker's own mother, is the one that does not do legitimate work. The comparison would not work if the ἀργίας γραφή were at this point a question of working your agricultural holdings. And the way the crime and its punishment are phrased in our sources, which only refer to it obliquely and not by quotation, huge swathes of the Classical citizen population, much of which was unlanded, would have suffered ἀτιμία if cultivation of owned agricultural property were necessary by the law. This is of course not the case: the Athenian Empire was driven by its landless, who enabled the existence of the fleet. But what exactly the ἀργίας γραφή, if indeed at some point it was expanded to include non-agricultural labor or this provision was included from the start, is looking for, then, is a little bit of a mystery. I cannot support the belief that the ἀργίας γραφή was a provision for some sort of "regular" work. That doesn't make any sense, since at Athens as at Rome and elsewhere large groups of the landless population must have been engaged in non-skilled labor, and therefore were without a "trade" or a steady source of work. I'm not even sure if that even works with the assumption that ἀργίας γραφή is looking for agricultural labor--you're not "on the clock" working your own fields.

In any case, the actual procedure presents its own problems, but not really the ones you want. We're not entirely sure what court the ἀργίας γραφή would have been taken up in, but we do know that in the Classical Period the crime was the result of public accusation, meaning that any citizen could bring an accusation publicly about any other. Plutarch's remark that the Areopagus was supposed to examine every citizens' livelihood may have been true in Solon's time, but is unbelievable in the Classical Period. Probably, as in other similar types of public accusations, the archon first was responsible for determining whether there was sufficient uncertainty to bring the case to court, by a sort of hearing consisting of a bunch of questions asked to the defendant (ἀνάκρισις). Schmitz mentions a fragment of a comedy of Diphilus that says that at Corinth, where the comedy takes place, if anyone seems to be living lavishly it's normal to make an inquiry as to how exactly he gets that money--if he's spending more than he'd normally make, he must be getting that money criminally. I'm not totally convinced that this has anything to do with Athenian procedure. The comedy takes place at Corinth, and though New Comedy often mixes Athenian practice into foreign states the procedure outlined doesn't seem to refer to any of the usual practices of Athenian court procedure. Nonetheless, the fragment does list the sorts of people who do not have legitimate livelihoods, and we seem to be on better ground to figure that this is applicable to Athens (or the whole joke is lost). The list includes thieves, sycophants, and false witnesses (like Eubulides). The idea, then, in these court cases seems not to be to prove that the defendant earns money somehow, to to prove or disprove that this money is gotten by legitimate means. That makes a lot more sense--there is no suggestion in Athenian law that not making money is a crime (how would sons in their fathers' households avoid prosecution?), and the existence of unskilled laborers renders the idea of steady income from a single source impossible.

As for "proof" of the means by which the defendant was acquiring money, I don't think there's actually very much to say. If we think Herodotus is right and there was a procedure by which citizens declared the sources of their incomes, then it was necessary to disprove them, which would have been extremely difficult. But it's quite clear that there were public trials and that public accusations could be made regarding the ἀργίας γραφή. We do not have any references to any such trial ever taking place, however, so we cannot know what the general line of argument was. Nonetheless, in similar cases (e.g. accusations of being a non-citizen), the procedure was typically pretty simple. Proof rested typically on the presence of dependable witnesses--Athenian jurists went to great lengths to try to show or dismantle through questioning the reliability of witnesses. Enough reliable witnesses who could show that the defendant's story was more likely to be accurate than the prosecution's, and demonstration that the prosecution's story was logically inconsistent or somehow otherwise unsound, would have been a good start, provided that the prosecution wasn't an exceptionally good and persuasive orator (similarly, arguments against him had to be persuasive). That's not really so different than modern courts, despite the more advanced forensic science available and the actual existence of investigative police in the modern world, of which there was no concept in antiquity. It's likely, though with no record of any such trial taking place, that evidence such as ledgers, would have found a place in trials of ἀργίας γραφή, but of course this alone even in a modern court does not necessarily constitute proof, if the prosecution can reasonably argue that there might be reason that those documents are not real.