What year is my 350 with numbers V0817TBC and casting numbers GM 3970010?

The numbers you provided suggest that your Chevrolet 350 engine was manufactured in 1980. The casting number GM 3970010 corresponds to a 350 cubic inch engine produced between 1969 and 1980, with the suffix code V0817TBC indicating that it was assembled in August 1977 at the Van Nuys plant. Therefore, your engine most likely dates back to 1980.