If you touch a Najis item e.g. a dog, with wetness, then your touching part of your body or cloth becomes Najis and needs to be washed with water to be Taahir. If you touch a dog or any Najis item with out wetness, then your body of cloth is not Najis.
For health precaution, you are advised not to touch a dog even with dry hands because of the possibility of transfer of microbes or parasites from the dog to your body. You can read on the medical researches published online about the dangerous illnesses which can come to humans from dogs. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5319273/#:~:text=Viral%20infections%20such%20as%20rabies,staphylococcus%20aureus%20are%20the%20most