Complexity of covering subset of the monoid $(\{0,1\}^n, \text{OR})$
L. J. Stockmeyer proves that SET BASIS is $\mathrm{NP}$-complete.

Reference:
L.J. Stockmeyer, The set basis problem is $\mathrm{NP}$-complete, Tech. Report RC-5431, IBM, 1975

Each vector of your problem can be seen as subset of $\{1, 2, \dots, n\}$. And the component-wise OR operation is corresponding to set union. So a basis for a collection of Boolean vector in your problem is exactly a set basis.