In Koblitz, Introduction to Elliptic Curves and Modular Forms on page 188 it is defined $$\left( \frac{-1}{j}\right)=0$$ in case $j$ is even. Apart from that definition $\left( \frac{c}{d}\right)$ is pretty much defined like the Kronecker symbol. The Kronecker symbol is $0$ if and only if $c$ and $d$ have a common factor, so in particular $\left( \frac{-1}{j}\right)\ne 0$.
Can you explain why Koblitz is defining it a little bit different? Is that on purpose or a mistake?
What is the exact relation between the Kronecker symbol and Koblitz symbol? In which cases is which definition useful?
There exist various definitions of the Kronecker symbol. I prefer to view the Kronecker symbol as giving the splitting behaviour of a prime $p$ in a quadratic number field with discriminant $\Delta$, and thus define $(\Delta/p)$ only for discriminants $\Delta$. This is what Koblitz is doing; in his case, $\Delta = -4$, hence $(-1/p) = (-4/p)$ should be $0$ when $p$ and $\Delta$ have a factor in common.