The notion of natural density gives a suficient condition. Namely, one can prove that if $A \subseteq \mathbb{N}$ has positive upper n. density then $\sum_{a \in A} \frac{1}{a}$ diverges. This condition is not a necessary one, though. A well-know well-known result in Number Theory ascertains that the series of the reciprocals of the prime numbers diverges whereas $\pi(n) = o(n)$.
In the following list you are to encounter some previous discussions on MO that are closely related to the current inquiry of yours:
[1] http://mathoverflow.net/questions/13230/erdos-conjecture-on-arithmetic-progressions
[2] http://mathoverflow.net/questions/4596/on-the-series-1-2-1-3-1-5-1-7-1-11
References
I. Erdös's brilliant proof of the divergence of $\sum_{p} \frac{1}{p}$: http://www.renyi.hu/~p_erdos/1938-13.pdf
