The Constitution does not apportion Congressional districts or presidential electors among the state based upon the number of citizens. It apportions districts and electors based upon the total population:
Representatives and direct Taxes shall be apportioned among the several States which may be included within this Union, according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons.Article I, Section 2. "Free persons" includes immigrants, both legal and illegal but no State allows non-citizens to vote.
In other words, immigration increases the relative power of high-immigrant states without changing the electorate. With the exception of California, the states favored by immigrants, particularly illegal immigrants, are Republican states. Moreover, although illegal immigrants count towards apportionment now, counting people legally in the country is much easier and more accurate than counting illegals. So legalization, greater immigration and even continued illegal immigration will necessarily increase Republican power in the federal government in the short term.
The medium term point is more subtle and less certain. Even after newly legalized immigrants become citizens and they and their children start to vote, the Republicans can still benefit even if they tend to vote Democratic. (Obviously, the Republicans benefit if the new citizens vote Republican.)
In the 2004 election, a historically high 44% of Latinos voted for George Bush. (This number is somewhat controversial, but is the best number available.) The Latino split between the parties is all over the place over time, but the Latino Republican vote varies between 20% and 40%. (Interestingly, the two Republican nominees who did best among Latinos are the two most conservative modern candidates, Ronald Reagan and George W. Bush.) People look at these numbers and think that this means that Latino immigration is great for Democrats. But that's because they don't think it through.
For purposes of this example, I'm going to look at the Texas vote in 2004 and assume that a Republican nominee with any chance at all will receive 40% of the Texan Latino vote. The point I'm making, however, doesn't depend upon the exact percentages and the following calculations can be run for any state and any percentage without changing the point. In 2004, George Bush got approximately 60% of the Texas vote. W beat John Kerry in Texas by 1.7 million votes and, because electoral votes in most states are winner take all, W got all of Texas' 34 electoral votes. Even if there had been 1.5 million more Latino Kerry votes, W still would have won the state and all of its electors. In fact, he would have won it handily, because 1.5 million more Kerry votes implies (assuming our 60/40 split) 1 million more Bush voters and a margin of 1.2 million votes. It's not until we add 7 million more Latino voters, 4.2 million voting for Kerry and 2.8 million voting for Bush, that Kerry even comes close to catching Bush. Very conservatively, 7 million more Latino voters implies 12-15 million new Latino citizens. Now, remember that the states currently get a new Congressional district and a new elector for every 650,000 people in the state. Suddenly, George Bush is not getting 34 electors for having won 60% of the Texas vote. He's getting 50-ish electors for having won just over 50% of the Texas vote. Moreover, because the vote for Representatives is also winner-take-all within each district, it is a good bet that most of the new Congressional districts also have a Republican congressman. If the Republicans can pick up even two or three net Congressional seats in each of the border states, they will have a lock on the House for the foreseeable future.
This specific example is ludicrously favorable to the Republicans. At this point, we've more or less doubled the Texas electorate and increased the population by 50%. George Bush is a Texas native son and did better among Latinos than any other Republican is likely to do for a while. But the general point should be clear: because of the way Congressional districts are apportioned; because of winner-take-all; because the Republican party is strong in the border states, immigration will aid Republicans in the short term and probably in the medium term even if new Latino citizens vote strongly Democratic.
In the long term, of course, we're all dead.
1. I've run the numbers for hypothetical state "T" with a population of 23,000,000, a voter to population rate of .35 that is consistent over all population sub-groups, a current Republican/Democrat split of 55/45 and a new Latino voter Republican/Democrat split of 20/80. In T, the Republicans will still squeak out a victory with an additional 1.3 million Latino voters. That implies a population increase of 3.7 million Latino "persons" and, thus, five additional Congressional districts and electors.
2. Given the country's overall demographics and given that Congressional apportionment is a zero sum game, the new Southern/Southwestern districts will come at the expense of Northern/Northeastern districts. In other words, each new district Republicans gain from immigration will most likely come at the cost of a Democratic district. Even if Republicans don't gain every new seat and Democrats don't lose every old seat, a shift from the North/Northeast to the South/Southwest would likely be a shift from a more liberal Representative to a more conservative Representative, moving the center of Congressional power to the right.