A lex Claudia in 218 prohibited senators from owning ships beyond a certain size, i.e. banning them from owning trade ships. Censors also could place a nota on a senator, or even remove him from the senate, for various financial misdemeanors, which might include the improper accumulation of income and which would certainly (although censors had great freedom to make their own interpretations for the purpose of notae) include something like running a market stall.

So one possible way to answer this question is to say that senators did often engage in commerce. The lex Claudia forbade them from owning their own trade ships, but there was no reason why they could not fund an enterprise for monetary return or otherwise make arrangements related to nautical trade. In particular the lex Claudia did not prohibit senators from placing their own freedmen or slaves in charge of ships or shops as part of their peculium, a sort of "administration" of assets that legally speaking were the not the property of the person who administered them (except in certain legal actions). We know that people did this pretty frequently. There was no mechanism by which a senator would be ejected automatically from his position for violating any of this. As with all Roman statutes, for the lex Claudia to have any force somebody would have to bring suit, and if the accused senator was found guilty he'd lose the ship. And for him to be ejected for financial reasons the censors would actually have to do that--keep in mind that for most of the first century there were no censors, or those that existed did not complete their assignments. Evidently people were being ejected from the senate occasionally--Lentulus, the highest-ranking of the Catilinarian conspirators, was cos. 71, expelled from the senate by the censors of 70, and then rose back up through the ranks again to become praetor in 63, when he was executed--but it was rare. Lentulus had been removed not as part of the normal operation of the censorship, but because in 70 a purge of the senate was conducted, to eliminate the traces of Sulla. So it was entirely possible for a senator to make money from business, and we're pretty confident that many did.

Another possible answer to this is that senators made money from land. A censor conducting a review of the senators expected senators to say that they were getting their income from the produce of their land and from rents. Senators were major landholders, not only holding estates of their own--which, like most Italian estates, were mostly simultaneously pleasure villas and farms--but also holding properties, typically in the city, for rent. Cicero's letters reveal to us the complex land-trading in which all senators were constantly involved--Cicero was wealthy, but not fabulously so. To a very real degree senators were wealthy on paper, but not necessarily in fact, because their land was not easily liquidated.

This last is an important point, because perhaps the best answer to how senators became the richest Romans is that they were not the richest Romans. They could be, but were not necessarily. A senator's paper wealth was typically pretty good--while there was no actual wealth requirement for the senate until Augustus, there was a wealth requirement to campaign for the quaestorship, which effectively imposed an unstated wealth requirement on senatorial membership. But his actual access to cash assets could be pretty poor. Political office could be both a money sink but also a source of liquid assets--military command in particular offered extortion of provinces and plunder. Nonetheless, political office was typically considered a major money sink, as was the senatorial lifestyle, which was highly competitive and very public, requiring large amounts of cash to keep up. Almost stereotypically the wealthiest equites were often far wealthier than senators, a fact that many authors of senatorial rank decried. So Maecenas, the second most powerful person in Augustus' regime and the financier of much of the Augustan cultural program, was just an eques. For senators in particular there was a disconnect between their "true," landed wealth and their available cash. As Frederiksen showed way back in 1966, senators were constantly in debt. The cost of existing as a senator, with all that that implied, vastly outdid the immediate cash resources of all but the smallest fraction, if even that, of senators. And their debts could, and often did, exceed the cash value of their landed wealth. Cicero joked only a year after putting down the Catilinarian conspiracy, a big point of which had been erasure of debts, that the purchase of his famous Palatine house had cost so much that he would be willing to join any revolution to deal with the debt. Debt was so prevalent among senators, who were constantly borrowing money from each other, often using other people's credit to back up their requests, that during the cash crisis at the beginning of the civil war Caesar allowed people to trade IOUs at face value, as if they were bank notes.