1. Let the oxidation state of chromium in $[Cr(NH_3)_4Cl_2]^+$ be $x$.

2. Determine the charge contributed by each ligand:

   • $NH_3$ is a neutral ligand, so each $NH_3$ contributes $0$.
   • $Cl^-$ is an anionic ligand, so each $Cl$ contributes $-1$.

3. The overall charge on the complex is $+1$. Therefore, $x + 4(0) + 2(-1) = +1$

4. Simplify: $x - 2 = 1$ $x = 3$

5. Hence, the oxidation state of Cr is: $\boxed{+3}$

6. Checking options:

   • A: $+3$ ✅
   • B: $+2$ ❌
   • C: $+1$ ❌
   • D: $0$ ❌

Therefore, the correct option is A.