I don't see how this would make money laundering for organized crime any easier than it is today, the tax would just be shifted to the landlord side (likely at a higher rate since they're probably in a higher tax bracket) and off the tenant.

Right now the tenant earns money, pays income tax on that money, pays rent, and the landlord pays taxes on that money (if they're honest and report it all) but can claim their mortgage interest as a tax deduction.

I think the tenant should be able to claim some portion of their rent as a tax deduction. It would require an official record of rent paid, which would keep the landlord honest. I'd say the mortgage interest on a rental property probably shouldn't be tax deductible either, but even still this would have the biggest impact on those large private landlords that are often what you'd call slumlords.

Edit: I'm obviously not an expert on taxation or housing policy so if I'm wildly out of touch I'll accept that, I just think it's kind of bullshit that the government subsidizes the mortgages we pay for our landlords with the money we paid the government when we worked for it.