Berndt showed that the number of zeros of $\zeta^{(k)}(s)$ for $0<t<T$ is
$$
N_k(T)=\frac{T}{2\pi}\left(\log\left(\frac{T}{4\pi}\right)-1\right)+O\left(\log T\right).
$$
From this I think it follows that $\zeta^{(k)}(s)$ has the same [genus][1] and hence the same order as $\zeta(s)$.

[1]: https://en.wikipedia.org/wiki/Entire_function#Genus_of_an_entire_function