The notion of tax-driven money is easiest to understand in relation to an exogenous tax such as a property tax or simple head tax. Demand for a state money is most effectively driven by exogenous taxes, not endogenous ones such as income taxes. Even so, in a hypothetical system with a tax imposed solely on income, the tax would still drive demand for a state money. It is worth considering why this is the case, because it also indicates why some level (though not absolutely any level) of basic income would also be consistent with currency viability.
Consider, first, an exogenous tax. The imposition of a simple head tax drives demand for a currency by ensuring that some people need to acquire income to pay the tax. This enables the government to induce labor services and also creates a supply of workers willing to work for the currency in the private sector. Anyone lacking sufficient savings will have a need, as a result of a head tax, to acquire income to meet the tax obligation.
An income tax is somewhat different. Those without an income need not pay the tax. The imposition of an income tax would not be sufficient to induce labor services if people were able and content to live on air. Of course, since people are not able or content to live on air, an income tax will drive demand for a currency much like a property tax or head tax. In other words, an income tax can drive demand for a state money because of the need or desire of many to obtain an income.
Private ownership of the means of production also ensures a supply of ready workers, but this is a willingness of workers to supply labor services without regard to the particular money that is paid. In principle, a private money could take the place of the state money. The tax imposed by the government ensures a supply of labor services in exchange for the state money in particular.
Now, the introduction of basic income would obviously reduce to some extent the capacity of an income tax (or other tax) to induce labor services. People cannot live on air, but they could choose to live on basic income, provided the level of the payment was at least sufficient for subsistence. It is possible, then, that basic income set high enough could completely destroy the capacity of an income tax (or other tax) to induce labor services. This would occur if everybody was content to live on basic income.
But if, after the introduction of basic income, some people still wanted more, the capacity of an income tax to drive demand for the currency would remain operative to some extent.
And if not just some, but most, people wanted more than the basic income, the capacity of an income tax to drive demand for the currency might be left largely unaffected.
This is especially so once non-financial factors are taken into account, such as the social benefits or sense of purpose that many people derive from their jobs. The empirical finding of the Modern Monetary Theorists in relation to Argentina, in which the poor claim to prefer a job to an income without a job, not only makes a strong case for a job guarantee but illustrates the likelihood that basic income would not seriously undermine the capacity of a tax to induce labor services. It would, however, give workers the power to say no to undesirable jobs, which would be a good thing, and put them on a more equal footing with those possessing independent means and facing no economic compulsion to join the labor force.
In any case, clearly there is some level of basic income that would not seriously impact the viability of the currency or make it impossible to induce labor services. Just as most cannot do without an income, most desire a much higher income than would be provided under any realistic level of basic income.