I suspect the law-and-order response to the policy in San Diego would be something along the lines of "if they can’t prove they’re clean, they don’t deserve my tax dollars." Of course, if everyone who received any sort of government assistance had to consent to a search of their home, the Fourth Amendment would be pretty much null(er). For example, I’d guess there’d be quite a bit more outrage if these fishing expeditions/searches were being done on the homes of, say, middle class kids applying for government-subsidized student loans instead of low-income people applying for welfare.
There might be more outrage, but of course many of the same justifications for searching the homes of welfare recipients might apply to student loan recipients. In any case, this practice points to one of the often-unappreciated costs of the massive welfare state: it inevitably leads to increased government monitoring of the beneficiaries of welfare programs. And of course as people become accustomed to searches of all sorts, the willingness to object or resist declines.
The power to do something for you is also the power to do something to you.