No food stamps, no welfare, no SSDI for illegals. The entire cost of each illegal picked up by state/local taxpayers.
Potential insolvency, degraded bond ratings, etc. will force the change.
There are USSC rulings that make it difficult to coerce the states by threatening their current funding. I am not a lawyer and I may be wrong about where the legal boundaries lie, but the main legal way Congress can act is on issues involving future funding. For example if Congress wanted to encourage voter reform and provided say $100 Billion to cover the cost, then they could set conditions to qualify. But they generally cannot say “if you don’t detain suspects for ICE we will cut off funding for XYZ”
Unless of course, the funds already had set conditions. Then if a state refuses to abide they can lose funding.
Another way they may be able do it is to change the law as it applies to all states.. This may be politically acceptable, too, since the states that voted for Hillary are the ones with the majority of immigration. So such a move may work since the GOP may not get much political blowback. Just speaking in general terms for illustration, Congress could repeal programs for social welfare and replace them with new programs that are contngent on certain conditions. If they tried this, it will be one heck of a fight with tons of protest.