They can have an effective strategy if they invoke the economy every single time. Economic factors used to be the Republicans greatest strength (until the debt Congresses of 2000-2006).
They need to state always and continuously that any vote on immigration reform while millions of Americans are out of work, especially blue collar workers, would hurt them particularly, and the government should first and foremost work towards getting the economy back on its feet. That means ending regulations, laws and tax policies that hurt US businesses. Mention that if the democrats really cared about the unemployed, rather than extending their benefits a few more weeks they should join the Republicans in making jobs job 1 in the government!
And the Reps can make that point citing the horrendous unemployment figures among blacks.
I totally agree with that and was going to post something similar upthread but I got sidetracked. I’ve heard a few say it, but not many.