How does the reciprocal lattice takes into account the basis of a crystal structure?

Your understanding that the reciprocal lattice is only defined for a Bravais lattice is correct. When you have a so-called lattice with a basis, you would calculate the reciprocal lattice primitive vectors using only the primitive vectors of the underlying Bravais lattice as:

\begin{aligned}\mathbf {b} _{1}&=2\pi {\frac {\mathbf {a} _{2}\times \mathbf {a} _{3}}{\mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)}}\\\mathbf {b} _{2}&=2\pi {\frac {\mathbf {a} _{3}\times \mathbf {a} _{1}}{\mathbf {a} _{2}\cdot \left(\mathbf {a} _{3}\times \mathbf {a} _{1}\right)}}\\\mathbf {b} _{3}&=2\pi {\frac {\mathbf {a} _{1}\times \mathbf {a} _{2}}{\mathbf {a} _{3}\cdot \left(\mathbf {a} _{1}\times \mathbf {a} _{2}\right)}}\end{aligned}

Indeed, this does not take into account the effect of the basis. That, however, plays a role in X-ray diffraction experiments and affects which peaks are actually measured. The resultant intensity is determined by how the waves diffracted from each of the constituent atoms of the basis interfere. For instance, consider the body-centered cubic lattice as a simple cubic lattice with a basis at $(0,0)$ and $(\frac{a}{2}, \frac{a}{2}, \frac{a}{2})$. It's possible that at some incident angles diffracted waves from these two atoms interfere destructively and you don't see an intensity peak even though you would expect it from a simple cubic reciprocal lattice. This is taken into account by the geometrical structure factor $F_{hkl}$, defined as: $$F_{hk\ell }=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}$$
where $(hkl)$ indicates the scattering plane, the index $j$ is summed over each atom in the basis with coordinates $(x_{j},y_{j},z_{j})$ and scattering factor $f_{j}$. This sum can be zero for certain values for $h,k,l$ and in such cases, no diffraction peak is seen even though it is expected from the reciprocal lattice structure of the underlying Bravais lattice.

(I've ignored that BCC is itself a Bravais lattice, for the purpose of this example. Of course, you could calculate the BCC reciprocal lattice primitive vectors using BCC primitive vectors and you'd get the right reciprocal lattice. XRD peak intensities thus determined will be most accurate, but simple structure factor calculations for SC with a basis would at least tell you which peaks won't be observed at all.)

Chapter 6 of the book Solid State Physics by Ashcroft and Mermin is a good reference for X-ray diffraction, and the concept of structure factor in particular.