Well, there is NO absolutely safe way to take photographs of stags during the rutting season safely. Deer are wild animals and as such totally unpredictable so if you want to be totally safe, don't go anywhere near them. But I know that some people (myself included) love to photograph these wonderful animals and want to minimise the risk of getting hurt or even killed.
I am a semi professional photographer.
I sell photos, undertake commissions and teach photography and post processing.
Photos that I have taken of stags have usually been taken in Royal Parks and I do not have permission to sell them (The Crown owns the Royal Parks). So, most of the photographs that I have taken of stag deer have not been taken with a view to selling them.