deposition testimony use in another case

What is the name of your state? Minnesota

I am being deposed in a civil matter, I know that during depositions attorneys have a lot of leeway in asking questions, including questions that one might consider irrelevent. If, during the deposition, the attorney were to ask a question of me, such as, for example, have I ever illegally downloaded a certain artist's music (this would not be related to the current case, but the person suing me does know I have illegally downloaded music in the past), and I answer truthfully that I have, can the copyright owner of the music then use this deposition testimony to sue me? Are my answers during the deposition public for anyone to use against me at a later time?

Even in federal court (which tends to be less limiting on deposition testimony than state), the Fifth Amendment is a valid privilege. Be sure not answering will not hurt you in the ability to respond to facts they may prove.

Unfortunately I don't think the 5th would work since small time copyright infringement is usually not a criminal offense.

Hoffman v. United States (1951) 341 U.S. 479
The privilege can be invoked whenever the claimant has a reasonable apprehension that an answer might tend to incriminate him. The privilege forbids compelled disclosures which could serve as a "link in a chain" of evidence tending to establish criminal liablility.

Obviously, the specific facts are very important. Did you get the item on a p2p site and put it in a shared directory? Did you download from a foreign country where it would be under a more specific law regarding the download? How many times are we talking about here? Many others.

Don't answer, use the privilege. If they make a motion to compel, make the decision to fight it (do you have a reasonable apprehension?) or comply, based on your risk/reward calculation of future lawsuit or current attorney fees.