If I take my computer that I use for shopping, personal banking and bill paying to a shop to be repaired, does the technician have access to my credit cards, bank accounts and passwords? Would it make sense to wipe my hard drive with one of those DOD programs?

Can they? Yes. Depending on how secure your data is anything stoed on your system is vulnerable.

However the likelihood of them doing so is very small. How long do you think they would stay in business if people suspected them of stealing personal info?

Yes of course they could, and a good hacker can often do it setting 1000 miles away if you are online.

The one important factor is to find and use a shop that you know and trust