Ilya Somin on the Volokh Conspiracy says that it is inconsistent to be opposed to affirmative action but in favor of racial profiling. This is one of those surface inconsistencies that only appears if you make shallow assumptions about the reasoning behind the positions. There is no inconsistency in the conservative position when you go beyond the shallow description of the policy preferences and get to the underlying reasoning. Conservatives believe that the government should be a neutral referee in society, not only in race but in religion, in wealth, in employer/employee relations, in renter/landlord relations, in buyer/seller relations, and in general (there are other areas where conservatives do not believe that the government should be neutral, but those are each individual issues).
On on the issue of race, when the government supports affirmative action it is not being neutral. It is taking sides with one group of Americans against another group of Americans. In the conservative view, affirmative action is morally wrong for much the same reasons that union-shop laws or rent-control laws are morally wrong (independently of any economic effects) and that reason is that the government is unfairly taking sides between its citizens.
By contrast, racial profiling is not an example of the government taking sides with one race against another or one religions against another. Airline security is supposed to protect all Americans of all races and religions equally. In order to do execute this task as effectively and as efficiently as possible, with the least overall burden to the public, the government should use all of the information it has available on who is likely to be a threat. If that information includes racial and religious indicators, then the government should use that information. It is not the government which is taking sides; it is nature that is taking sides. The state of the world is such that the best security practices involve profiling Muslims and people from the Middle East. For the government to take cognizance of this fact is not racist, not prejudiced, but merely pragmatic.
It is much like the alleged government failure during the Katrina disaster. This supposed failure of the federal government had a disparate effect on black people but only someone who is conspiracy-minded and creepily obsessed with race would really believe that the federal government actually intended to hurt black people. The reason that black people were disparately harmed is because more black people lived in the area. This racially disparate result was caused by brute fact, not by government choice.
It is the same in the immigration area. The purpose of immigration law is to benefit all Americans and legal American immigrants. If enforcing this law effectively has disparate consequences on one particular race, this is not the choice of the government; it is not the government that has decided to take the side of one race over another; this is just what the facts on the ground require as a form of enforcement.
The conservative position is consistent here. The government should not takes sides of one race against another, but if the government in the course of carrying out its proper functions is required to do things that happen to effect people of one race more than another, it is no concern of the government to take this into account. In all cases, the government should ignore race as a matter of government concern, and but government actors may take account of race when necessary to do their race-neutral job.