Here's a proof of the statement for $f=0$ (so that $X=W$ is a Wiener process). As the terms involving $f$ remain well-behaved for any continuous $\phi$, I think that it should be possible to extend the usual proofs to include nonzero $f$ and arbitrary $\phi$. I'll base the proof on the following simple result. Here, I am using $B_\epsilon=\left\{\omega\in\mathcal{W}^n\colon\sup_{t\in[0,1]}\lVert\omega(t)\rVert\le\epsilon\right\}$ for the $\epsilon$-ball in Wiener space.

> **Lemma**: For any continuous $\gamma\colon[0,1]\to\mathbb{R}^n$ and $\epsilon > 0$
$$
\mathbb{P}\left(W\in B_\epsilon+\gamma\right)\le\mathbb{P}\left(W\in B_\epsilon\right).
$$

*Proof*: As the Wiener measure is Gaussian, it is [log-concave][1]. As $B_\epsilon$ is convex then its indicator function is log-concave and, as the convolution of log-concave functions is log-concave, this implies that $\gamma\mapsto\mathbb{P}(W\in B_\epsilon+\gamma)$ is log-concave. As it is also symmetric this gives
$$
\mathbb{P}\left(W\in B_\epsilon\right)\ge\sqrt{\mathbb{P}\left(W\in B_\epsilon+\gamma\right)\mathbb{P}\left(W\in B_\epsilon-\gamma\right)}=\mathbb{P}\left(W\in B_\epsilon+\gamma\right).
$$
*QED*

Choosing any smooth $\gamma\colon[0,1]\to\mathbb{R}^n$ with $\gamma(0)=0$ then, using a Girsanov transform in the usual way, 
$$
\begin{align}
&\mathbb{P}\left(W\in B_\epsilon+\phi\right)
=
\mathbb{P}\left(W+\gamma\in B_\epsilon+\phi+\gamma\right)\\
&\qquad=\mathbb{E}\left[\exp\left(\int_0^1\dot\gamma dW-\frac12\int_0^1\dot\gamma^2dt\right)\Bigg\vert W\in B_\epsilon+\phi+\gamma\right]\mathbb{P}(W\in B_\epsilon+\phi+\gamma)\\
&\qquad\le\mathbb{E}\left[\exp\left(\dot\gamma(1)W(1)-\int_0^1\ddot\gamma W dt-\frac12\int_0^1\dot\gamma^2dt\right)\Bigg\vert W\in B_\epsilon+\phi+\gamma\right]\mathbb{P}(W\in B_\epsilon)
\end{align}
$$
The last inequality here is using the lemma, together with integration by parts in the exponent. So, letting $\epsilon$ go to zero and using uniform convergence in the expectation,
$$
\begin{align}
\limsup_{\epsilon\to0}\frac{\mathbb{P}(W\in B_\epsilon+\phi)}{\mathbb{P}(W\in B_\epsilon)}
&\le\exp\left(\dot\gamma(1)(\phi(1)+\gamma(1))-\int_0^1\ddot\gamma (\phi+\gamma) dt-\frac12\int_0^1\dot\gamma^2dt\right)\\
&=\exp\left(\frac12\int_0^1\dot\gamma^2dt+\int_0^1\dot\gamma\dot\phi dt\right).
\end{align}
$$
In the last line, the derivative $\dot\phi$ is understood in the sense of distributions. Now, if $\phi$ is not in Cameron--Martin space, then if we let $\gamma$ be smooth approximations to $-\phi$ (e.g., convolve with a smooth bump function), the right hand side tends to zero giving the result.

  [1]: http://en.wikipedia.org/wiki/Logarithmically_concave_measure