The maximum value is $1004$.
Note that $\sin \theta_j \cos \theta_{j+1} \le \frac{1}{2}(\sin^2 \theta_j + \cos^2 \theta_{j+1})$ for any $j$, where the indices are taken modulo $2008$. It follows that
$$\sum_{j=1}^{2008} \sin \theta_j \cos \theta_{j+1} \le \frac{1}{2} \sum_{j=1}^{2008} (\sin^2 \theta_j + \cos^2 \theta_j) = \frac{1}{2} \sum_{j=1}^{2008} 1 = 1004.$$
Equality holds when $\theta_j = \frac{\pi}{4}$ for each $j$.