People who avoid taxes will still avoid taxes. Sure when they buy something legal they pay taxes, as they do today in the form of hidden taxes embedded in the products. But are the drug dealer and prostitute going to pay taxes on the goods and services they sell? See when you provide a $100 service or product to another person, you need to charge an extra $30 and remit it to the government. The illegal activities will still not do this and will cheat the system just as they do today. The fairtax does nothing to capture the underground economy despite their claims.
Those in this sort of illegal economy WILL, however, pay the full 23% ti FairTax rate on purchases of taxable items - just as will you and I (and at the same marginal rate which, again, is not the effective rate actually funded by the taxpayer).