Can basal carcinoma arise from existing mole that a person had for many years?

No but... It can arise adjacent to the nevus. Or the concerning lesion may be one of those unusual melanomas that arises in a mole.
Mole to BCC. Yes it is possible. Best to have a punch biopsy of the lesion for diagnosis.
May be, but. Any change in an existing mole needs to be taken seriously to rule out melanoma rather than presuming a basal cell carcinoma. It would be prudent to see your doctor for and examination.