$(X, \tau) $ be a topological space and suppose $\mathcal{B}$ is a basis of $X$ for the topology $\tau$.
It can be shown that if $\tau$ define a non discrete $T_1$ topology on $X$ , then $X$ can't have a minimal base for the topology $\tau$.i.e for any basis $\mathcal{B}$ there exists $\mathcal{B}'\subsetneq \mathcal{B}$ which is also a basis.
My Question : Suppose the topological space $(X, \tau) $ such that $X$ has no minimal base for the topology $\tau$ . Does this implies $(X, \tau) $ a non discrete $T_1$ space.
No, there are tons of non-$T_1$ topologies that have no minimal basis (and are not just a small modification of a $T_1$ space). For a simple example, take $\mathbb{R}$ with the topology consisting of the empty set and the sets $U_x=\{y\in\mathbb{R}:y>x\}$ for each $x\in\mathbb{R}$. This is very far from $T_1$ (no subspace with more than one point is $T_1$!) but I claim it has no minimal basis. Indeed, a collection of sets $U_x$ form a basis iff the corresponding $x$ values are dense in $\mathbb{R}$, and any dense subset of $\mathbb{R}$ remains dense if you remove an element.