What's the right approach to dismantling the black market? Liberalization. The immigration black market only exists is because the government has made the legal market as bersome as it can.
True immigration reform makes legal channels more appealing, not less. That means lightening the paperwork and the regulatory burden, and eliminating quotas. The more unattractive legality becomes, the more attractive illegality looks in comparison.
Black markets are anathema to a free society. Murder, theft, smuggling, and even slavery are part and parcel of immigrant black markets. They are also easily avoidable - just shrink the black market by making legal immigration easier.