You started out on the right foot using string, I'm not quite sure why you reverted to using character arrays and not using functions which are already provided for you in cctype.
Code:
string s;
int upper = 0, lower = 0;
char c;
cin >> s;
for (string::iterator i = s.begin(); i != s.end(); ++i)
{
          if (isupper(*i))
                    ++upper;
          else if (islower(*i))
                    ++lower;
}
cout << "upper: " << upper << '\n';
cout << "lower: " << lower << '\n';
cout << "convert to lowercase? ";
cin >> c
if (c == 'y' || c == 'Y')
          // convert string to lowercase