In Hirabayashi v. U.S., 320 U. S. 81, 1943, and Korematsu v. U.S., 323 U.S. 214 (1944), the Supreme Court found wide latitude in Congress's explicit Consititutional power "to make and declare war." From Hirabayashi: "The war power of the national government is 'the power to wage war successfully'."
There were three major Supreme Court cases regarding internment: Korematsu, Yasui and Hirabayashi. They challenged the exclusion order and curfews. The Supreme Court upheld the Japanese men's convictions and ultimately the government's power to intern Japanese-Americans.
In the 1980's lawyers on behalf of the Japanese men petitioned for writ of error coram nobis, a somewhat obscure procedure used to correct a fundamental error of fact in a trial after the defendant has been found guilty and served his sentence. They were offered a pardon, to which Korematsu famously replied, "We should be the ones pardoning the government." Hirabayashi and Koramatsu both got their day in Court and their convictions were eventually vacated. Unfortunately, Yasui died before he could appeal the District Court Judge's granting of the government's Motion to Dismiss.
Interment Victims also filed civil suits, but none that I'm aware of were successful. Eventually, Congress appropriated reparations money and victims were given $20,000.00.
Sources:
Korematsu v. United States: A Constant Caution in a Time of Crisis
If you're interested in more things related to Japanese Internment, check out http://densho.org - it's the clearinghouse for all things related to the Japanese American experience in WWII. It has interviews with survivors, primary source documentation, and more.