Besides criminal law, the courts also offer remedies through civil actions (lawsuits). The nature of lawsuits is such that you can basically sue anyone for any reasonable claim of damages or injury. You can even sue parties unknown, as long as some hope exists of identifying them at a later time. You can ask for actual and punitive payment of damages, injunctive relief (cease-and-desist orders), as well as other remedies for damage. Often, a case is easier to prove in a civil court than in a criminal court. Civil cases require a less strict standard of proof to convict than do criminal cases (a "preponderance of the evidence" as opposed to the "beyond a reasonable doubt" standard used in criminal cases).
Civil remedies are much easier to get after a criminal conviction is obtained.
However, filing civil suits for damages in the case of break-ins or unauthorized activity may not be a practical course of action for several reasons:
Civil suits can be very expensive to pursue. Criminal cases are prosecuted by a government unit, and evidence collection and trial preparation are paid for by the government. In a civil case, the gathering of evidence and the preparation of trial material is done by your lawyers - and you pay for it. Good lawyers still charge more than most computer consultants; therefore, the cost of preparing even a minor suit may be quite high. If you win your case, you may be able to have your lawyers' fees included in your award, but don't count on such provisions. Note, however, that criminal cases may sometimes be used as the basis for later civil action.
The preparation and scheduling of a lawsuit may take many years. In the meantime, unless you are able to get temporary injunctions, whatever behavior that you are seeking to punish may continue. The delay can also continue to increase the cost of your preparation.
You may not win a civil case. Even if you do win, it's possible that your opponent will not have any resources with which to pay damages or your lawyer fees, and so you therefore may be out a considerable sum of money without any satisfaction other than the moral victory of a court decision in your favor. What's more, your opponent can always appeal - or countersue.
The vast majority of civil cases are settled out of court. Simply beginning the process of bringing suit may be sufficient to force your opponent into a settlement. If all you are seeking is to stop someone from breaking into your system, or force someone to repay you for stolen computer time or documents, a lawsuit may be an appropriate course of action. In many cases, especially those involving juveniles trying to break into your system, having a lawyer send an official letter of complaint demanding an immediate stop to the activity may be a cheap and effective way of achieving your goal.
As with other legal matters, you should consult with a trained attorney to discuss trade-offs and alternatives. There is a wide range of possibilities available to you depending on your locale, the nature of your complaint, and the nature of your opponent (business, individual, etc.).