NOTE: I am not a lawyer, so check the links and do your own research before reaching any conclusions.
Basis for Federal criminal jurisdiction
Because the Constitution specifies that powers not specifically granted to the Federal government belong to the States, or the people, the scope of Federal law is supposedly limited. Unlike the States, the Federal government does not enjoy a general police power. Still, few Federal criminal statutes seem based on clearly articulated Constitutional provisions. Two exceptions are the income tax (Sixteenth Amendment) and the prohibition of counterfeiting (Article I, Section 8).
Most Federal criminal law is based on more-or-less loose interpretations of the Federal government´s power to levy taxes and to regulate interstate commerce (both in Article I, Section 8). Federal laws that prohibit certain drugs and make others available only by prescription, and laws that require the registration of machineguns, are based on the power to tax. Other Federal laws - for example, bank fraud, bank robbery and gun dealer licensing - are based on the power to regulate interstate commerce. Over the years, the Supreme Court has repeatedly upheld legislation that seems only marginally related to the Federal government´s powers to tax and regulate commerce. Interestingly, the Court has begun placing some limits, for example, by striking down the "gun-free school zone" act as being too marginal to commerce. But it is probably a fair thing to say that if Congress passes a Federal criminal law, the Court, in its deference to the will of the legislative, is likely to approve.
Constitution of the United States – Selected Articles and Amendments
(italics are mine)
We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this Constitution for the United States of America.
Section 8. The Congress shall have power to lay and collect taxes, duties, imposts and excises, to pay the debts and provide for the common defense and general welfare of the United States; but all duties, imposts and excises shall be uniform throughout the United States;
To borrow money on the credit of the United States;
To regulate commerce with foreign nations, and among the several states, and with the Indian tribes;
To establish a uniform rule of naturalization, and uniform laws on the subject of bankruptcies throughout the United States;
To coin money, regulate the value thereof, and of foreign coin, and fix the standard of weights and measures;
To provide for the punishment of counterfeiting the securities and current coin of the United States;
To establish post offices and post roads;
To promote the progress of science and useful arts, by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries;
To constitute tribunals inferior to the Supreme Court;
To define and punish piracies and felonies committed on the high seas, and offenses against the law of nations;
To declare war, grant letters of marque and reprisal, and make rules concerning captures on land and water;
To raise and support armies, but no appropriation of money to that use shall be for a longer term than two years;
To provide and maintain a navy;
To make rules for the government and regulation of the land and naval forces;
To provide for calling forth the militia to execute the laws of the union, suppress insurrections and repel invasions;
To provide for organizing, arming, and disciplining, the militia, and for governing such part of them as may be employed in the service of the United States, reserving to the states respectively, the appointment of the officers, and the authority of training the militia according to the discipline prescribed by Congress;
To exercise exclusive legislation in all cases whatsoever, over such District (not exceeding ten miles square) as may, by cession of particular states, and the acceptance of Congress, become the seat of the government of the United States, and to exercise like authority over all places purchased by the consent of the legislature of the state in which the same shall be, for the erection of forts, magazines, arsenals, dockyards, and other needful buildings;--And
To make all laws which shall be necessary and proper for carrying into execution the foregoing powers, and all other powers vested by this Constitution in the government of the United States, or in any department or officer thereof.
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
The Congress shall have power to lay and collect taxes on incomes, from whatever source derived, without apportionment among the several states, and without regard to any census or enumeration.