The only crime defined in the Constitution of the United States is treason. The reason why treason is defined is that, under English common law, crimes were defined by courts based on what they believed violated justice. Without proper definition, anyone could be sentenced for treason on a court's whim.
Treason is defined by the Constitution as the act of going to war against the United States or aiding its enemies. The Constitution also states that two witnesses are required to convict someone of treason. It also gives power to the Congress to set the punishment. Treason against the United States has been defined under the US Code - chapter 115.