RBC's in circulation do not have a nucleus. The nucleus is ejected as part of cell differentiation when the RBC is formed, long before it goes into body circulation. However white blood cells do have nuclei, so you can get DNA from them and use that for DNA fingerprinting. Because the number of white cells in a sample is relatively small, the DNA will need to be amplified before it can be run, but that is commonly available technology in a lab capable of DNA analysis.
Red blood cells do have nuclei, when they are newly created. So maybe it is new RBC's that are bing used to supply the DNA. They tend to lose those nuclei as they age, so maybe it is the new RBC's with nuclei that are used to provide genetic fingerprints.
As I understand it, you can use almost any blood sample to take a DNA finger print. If the sample is diluted or corrupted, you might not be able to use it. However, a person’s blood is generally going to have DNA no matter where it comes from.