Is it a bad idea to use a public defender to fight a DUI?

In my area (northern NJ), attorneys who do DUI work tend to charge around $3,000 or so, but there is wide variation, as there is in all attorney costs: you can find higher and lower. That said, a few thousand dollars--not a few hundred--is the norm. DUI cases can be very time consuming and expensive for lawyers, typically involving both multiple court appearances and lots of "paperwork": discovery (getting and reviewing information) and motion practice (formal requests to the court for the judge to do or order something). If you are in an area where a lawyer is less likely to have or be able to string together multiple cases for the same day, that will likely increase cost, since your case has to foot or support all the attorney's travel and court time each time he/she appears.
A public defender is typically experienced in court, but not necessarily with DUI, and is likely committed to the goal of justice--and is also overworked, underpaid, and under-resourced, and has no real incentive to do well in your case, other than any satistaction from doing the job well, since they draw the same (fairly small) salary win, lose or draw. A public defender is better than no attorney, but an experienced DUI lawyer who is hired by you specifically (not by the state) is better than a pubic defender.