Even numbers do not contribute to any odd coefficients, hence we have $$\begin{align*} &((x+1)(x+2)\cdots(x+2015))^{2016} \\ \sim\ & ((x+1)x(x+1)x\cdots (x+1))^{2016} \\ \sim\ & (x^{1007}\cdot (x+1)^{1008})^{2016} \\ \sim\ & (x+1)^{1008\times 2016} \end{align*}$$

Here, we use the $\sim$ operator to indicate the two expressions will have the same result.

Because the binary representation of $(1008 \times 2016)$ has six $1$'s, so there will be $2^6=\boxed{64}$ odd coefficients.