Answer: B

Solution:

Since $1 + 2 + \cdots + n = \dfrac{n(n+1)}{2},$ the quotient is $\dfrac{6n}{n(n+1)/2} = \dfrac{12}{n+1},$ which is an integer exactly when $n + 1$ divides $12.$ The divisors of $12$ that are at least $2$ are $2, 3, 4, 6, 12,$ giving $n = 1, 2, 3, 5, 11$ — five values.

Thus, the correct answer is B.