I have a birth mark that I'm afraid may have turned into basal cell carcinoma. Is this even possible?

Paul, the only right and smart thing to do is to visit your physician, preferably a skin doctor MD dermatologist whenever a long-existing "birth mark" begins to undergo change. While a basal cell carcinoma is almost never found on the penis, other possibilities include a melanoma or even a sexually transmitted infection or virus. You are not over-reacting. Now call for a doctor's visit this week.