When performing a DNA cloning, sometimes PCR amplicon is run in agarose and it is detected by ethidium bromide marking under UV light. After that, gel is sliced, DNA extracted from gel....... until eventually you have your transgenic organism. The thing is that when doing so, you use the very same DNA sample that was "contaminated" by ethidium bromide, so does this DNA you are using for transformation contain ethidium bromide? if so, how can it affect in numbers to mutation risk?
Purified DNA contains negligible amounts of ethidium bromide. PCR and gel clean-up kits remove it quite well. There is, though, a risk of mutation from the fact that you're visualizing the gel in UV light with ethidium bromide. The risk of mutation from UV is minimized by exposing the gel as little as possible, and by using "preparative" transilluminators that come with a low-light setting for cloning purposes. If you work well and quickly, the chance of mutagenesis isn't higher than the chance of a polymerase incorporating the wrong nucleotide.
While it is possible to get mutations from the ethidium bromide, this is hardly the only source of mutations in the process. Taq itself will introduce mutations fairly frequently, so sequence sensitive procedures typically use a higher accuracy (but slower and more expensive) polymerase, like Pfu. As stated before, the UV can also cause mutations. The PCR process iteself can result in primer dimers and other "funky business". Inappropriate sequences can be picked up from environmental contamination, particularly if you frequently use the same restriction enzymes in the lab, because sequences with appropriate "sticky ends" will be all over the place, no matter how much you clean. If inserting into a plasmid, multiple inserts can get ligated into the final product, depending on how homologous the sticky ends of the insert are.
The solution is to accurately sequence the final transgenic product before you put it in the transgenic organism. I usually have the plasmid (I work with plasmid transfections) sequenced, as well as cleaving the plasmid insert from the plasmid and running it on a gel. The sequencing catches any base swaps/indels and the gel makes sure there is only one copy of the insert in that particular plasmid.
You would be surprised how often you get mutations and other "garbage" in the final product, which is why it's important to make multiple batches together. I make mine in batches of 8, so that at least a couple will be exactly what I intended to produce.
Reference: My own research/experience with site-directed mutagenesis and construction of plasmid vectors for transfection.