No agency should be enforcing copyrights. It should be purely a matter of civil law (i.e. if you think somebody is infringing your copyright, sue them and collect damages if you prove your case).
As a policy matter, you’re right, copyright law should be purely civil law. However, Congress clearly has the Constitutional authority to enact criminal copyright laws (Art. 1, Sec. 8 says that Congress may “secur[e] for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries,” but says nothing about how Congress may “secure” copyrights). Since Congress has done so, I’m just wondering who is the “correct” enforcement agency.
A criminal conspiracy to violate copyright, however, should fall under the Department of Justice for the related conspiracy, racketeering, etc., laws.