The space-time interval in General Relativity is $ds^2=g_{\mu\nu}dx^\mu dx^\nu$, where $g_{\mu\nu}$ is the metric tensor. If the space-time is stationary and axisymmetric, the corresponding geometry is described by the Kerr metric, where the non-zero metric coefficients are $g_{tt}$, $g_{rr}$, $g_{\theta\theta}$, $g_{\phi\phi}$ and $g_{t\phi}$. The off-diagonal term $g_{t\phi}\ne0$ implies the axisymmetry.
In the textbook by Landau & Lifshitz, the authors define the 3-dimensional metric tensor $$\gamma_{ij}=\left(-g_{ij}+\frac{g_{0i}g_{0j}}{g_{00}}\right),$$ such that the spatial distance is expressed as $dl^2=\gamma_{ij}dx^idx^j$. Then, the authors define the 3-dimensional vector $\mathbf{g}$ as $$g_i=-\frac{g_{0i}}{g_{00}}.$$ I am trying to calculate the curl of the vector $\mathbf{g}$, i.e., $\mathbf{\nabla}\times\mathbf{g}$, and for this I require to choose a suitable set of basis vectors. However, I am not able to understand how to find the basis vectors.
Could someone please suggest me any hints on how to proceed with the calculation?
This might not be exactly, what you are looking for, but I am unsure, what you mean exactly by "finding basis vectors". If the task is to compute the curl, I suggest to proceed exactly as suggested by Ted Shifrin in the comments. Let me try to outline the concrete computation steps you need to do:
In the given situation, you are working on a 3-dimensional Riemannian manifold with metric $\gamma$. So the components of $\gamma$ can be expressed as $\gamma_{ij}$ with $i,j\in\{1,2,3\}$ in any given chart/coordinate system. So you can choose any coordinate expression for the Kerr metric you like and compute $\gamma$ using the formula from Landau/Lifshitz.
In the same way you can use the other formula from Landau/Lifshitz you gave to compute the components of $\mathbf{g}$. Note that although Landau/Lifshitz call this a vector, it is really a 1-form or a co-vector field, because you compute $g_i$ with a lower index. The components of the actual vector field (contra-variant components) can be computed by $g^i = \gamma^{ij}g_j$, where you need the contra-variant metric components $\gamma^{ij}$. Compute them, e.g., by computing the matrix inverse of $(\gamma_{ij})$.
On the Riemannian manifold with metric $\gamma$, we can define the curl of a vector field $X$ with components $X^i$ as $$ \nabla\times X := (\star_{\gamma}(\mathrm{d}X^\flat))^\sharp.$$ Here the musical isomorphisms $\flat$ and $\sharp$, just correspond to raising and lowering indices, i.e., $(X^\flat)_i = \gamma_{ij}X^j$ for a vector field $X$ and $(u^\sharp)^i = \gamma^{ij}u_j$ for a co-vector field or 1-form $u$. The operator $\mathrm{d}$ is the exterior derivative. For a 1-form $u$ the component expression for the resulting 2-form $\mathrm{d}u$ is just $$(\mathrm{d}u)_{ij} = \partial_iu_j - \partial_ju_i.$$ $\star_\gamma$ is the Hodge star operator for the metric $\gamma$. In the 3-dimensional case here, it converts a 2-form into a 1-form. You can compute it for the metric $\gamma$, e.g., using the formula in this answer. For the 2-form $\mathrm{d}u$ in the 3-dimensional case here, it breaks down to $$ (\star_\gamma(\mathrm{d}u))_k = \sqrt{\det(\gamma)}\gamma^{il}\gamma^{jm}\varepsilon_{klm}(\mathrm{d}u)_{ij}.$$
Putting everything together, what you might want to compute is: $$ (\nabla\times\mathbf{g})^k = \sqrt{\det(\gamma)}\gamma^{kr}\gamma^{il}\gamma^{jm}\varepsilon_{rlm}(\partial_ig_j - \partial_jg_i).$$
On a side note: I think what Landau/Lifshitz is doing there is called 3+1-splitting or lapse/shift-splitting in general relativity (cf. for example $\S$ 21.4 in Misner, Thorne, Wheeler: Gravitation). The Riemannian metric $\gamma$ will then only be independent of the time $t$ you split away if the spacetime is stationary, which of course is true for the Kerr metric.