There are two main constitutional issues that are linked to the sort of racial and ethnic profiling that has occurred in the US since the 9/11 attacks (and, of course, even before that).
First, there is the 4th Amendment issue. The 4th Amendment says that the government cannot conduct unreasonable searches and seizures. It says that the government must have probable cause to conduct searches and seizures. The practice of racial or ethnic profiling can be seen as a violation of this requirement. If the police start to subject a person to surveillance simply because of his or her race or ethnicity, they may be searching that person without probable cause. The fact that a person is of a given race or ethnicity surely cannot constitute probable cause to suspect them of a crime.
Second, there is the 14th Amendment issue. The relevant part of the 14th Amendment says that the government must extend the equal protection of the laws to all people. That means that the government must treat all people equally. When the government singles out people of a certain race or ethnicity and subjects them to increased surveillance and scrutiny, it is not treating them in the same way that it treats people from other groups. This, arguably, violates the 14th Amendment.