Is it enough to be "suspected" of terrorism, or should these suspects have the right to a fair trial before being detained indefinitely?

They have the right to fair trial because it states so in the 6th Amendment of the Bill of Rights in the U.S Constitution.

Any system of indefinite detention without trial for suspected terrorists would be unnecessary, unwise, and unconstitutional. Individuals who pose a genuine danger to the United States can be prosecuted in our existing courts under our existing laws. And any statute that purports to authorize the long-term detention of suspected terrorists on the basis of perceived future dangerousness rather than past criminal conduct will likely be invalidated by our federal courts. Finally, a system of detention without trial for terrorism suspects would rightly be viewed by Americans and throughout the world as an illegitimate continuation of the Guantánamo detention regime that so damaged our credibility as a nation that adheres to the Constitution and the rule of law.