I disagree with Jack's answer. The federal Fair Credit Reporting Act (FCRA) limits the information that may be contained in a report obtained from a consumer reporting agency (CRA). Most of the adverse information on a consumer report (which is defined to include more than credit reports -- employment background checks fall under the act, too) are limited to adverse actions that occurred within the last 7 years. Criminal convictions do not have a time limit, but pre conviction criminal information like arrests, filing of charges, warrants, etc are subject to that 7 year limit. When an employer or creditor intends to use a third party background check (like a credit report) in making its decision to hire you or extend you credit the creditor/employer must disclose that fact to the applicant and get written permission from the consumer to obtain that report. If the employer/creditor denies employment/credit in part based on an item in a consumer report, the employer/creditor must disclose that fact to you, give you a copy of the report, and a copy of your rights to contest the adverse item if it is not accurate. If a CRA, employer, or creditor violates the FCRA you may sue for either statutory damages of up to $1,000 per violation or your actual damages. In addition to the damages, you may be awarded your attorney's fees, too.
Reporting a warrant from 2001 on a consumer report to be used for employment purposes would almost certainly violate the FCRA since that adverse action was taken much longer than 7 years ago. Note that if charges were filed and then dismissed years later, it is the date the charges were filed that starts the 7 year time running, not the dismissal, per a recent federal appeals court decision. So getting a dismissal now cannot be reported on your CRA as the dismissal doesn't count.
You may want to see an attorney who litigates FCRA cases to see if you might have a case worth pursuing. Since you can get attorney's fees included in the award, it can be worthwhile to hire the attorney even to get a statutory damage award of $1,000. You just want to be sure that your case has a pretty good chance to succeed or you may be stuck paying attorney's fees yourself if you lose. Discuss with the attorney exactly how the fee agreement works if you win and if you lose so you are clear on what you might be liable for. You might want to talk to more than one attorney.
You'll also want to contact the CRA that provided the report to contest that old warrant information and ask that it be removed immediately.