1. Criterion for pH dependence
   A half-cell reaction is pH dependent if its electrode potential depends on $[H^+]$ (or pH). From the Nernst equation, this happens when $H^+$ or $OH^-$ appears in the half-reaction.

2. Check each half-cell

Option A: $Pt\,|\,Fe^{3+},Fe^{2+}$

The half-reaction is $Fe^{3+}+e^- \rightarrow Fe^{2+}$ Nernst equation: $E=E^\circ-\frac{0.0591}{1}\log\frac{[Fe^{2+}]}{[Fe^{3+}]}$ No $H^+$ or $OH^-$ appears.
So, this electrode is not pH dependent.

Option B: $MnO_4^-\,|\,Mn^{2+}$

In acidic medium, the half-reaction is $MnO_4^-+8H^+ +5e^- \rightarrow Mn^{2+}+4H_2O$ Nernst equation: $E=E^\circ-\frac{0.0591}{5}\log\frac{[Mn^{2+}]}{[MnO_4^-][H^+]^8}$ Since $[H^+]$ appears explicitly, the electrode potential depends on pH.
So, this electrode is pH dependent.

Option C: $Ag\,|\,AgCl\,|Cl^-$

The half-reaction is $AgCl(s)+e^- \rightarrow Ag(s)+Cl^-$ Nernst equation: $E=E^\circ-0.0591\log[Cl^-]$ No $H^+$ or $OH^-$ appears.
So, this electrode is not pH dependent.

Option D: $\frac12F_2\,|F^-$

The half-reaction is $\frac12F_2+e^- \rightarrow F^-$ Nernst equation: $E=E^\circ-0.0591\log\frac{[F^-]}{(P_{F_2})^{1/2}}$ No $H^+$ or $OH^-$ appears.
So, this electrode is not pH dependent.

3. Conclusion
   Only Option B has $H^+$ in the half-cell reaction, so only this half-cell is pH dependent.

$\boxed{\text{B}}$

4. Comparison with stored answer
   Stored correct answer: B
   Our derived answer: B
   So they agree.