This post does not answer the OP's question, but provides certain experimental findings, which might be of interest to the OP, but which need to be established rigorously.
Representing each $X_i$ as an affine transform of the Bernoulli random variable, Mathematica can be summoned to evaluate the expectation for low values of $n$, here is the result for $0\leqslant n \leqslant 16$:
Few observations are in order:
- Expectations $\mathbb{E}\left(\sup\limits_{0 \leqslant k \leqslant n}S_k\right)$ appear to follow a pattern $\frac{p}{1-2p} + \mathcal{o}\left(p^{\lfloor n/2 \rfloor}\right)$, which is in agreement with the result of @did that $\mathbb{E}(S^\ast) = \frac{p}{1-2p}$.
- Expectations appear (as suggested by Mathematica's FindSequenceFunction) to be solutions of rank $3$ linear inhomogeneous recurrence equation:
That is $(n+3) S_{n+3}^\ast - (n-4) S_{n+2}^\ast - 4 n p (1-p) S_{n+1}^\ast + 4(n+1) p(1-p) S^\ast_n = -p(1-2p)$
- The generating function for the sequence of expectations $S_n^\ast$, i.e. $$g(x) = \sum\limits_{n=0}^\infty x^{n} S_n^\ast = \frac{\sqrt{4 p^2 x^2-4 p x^2+1}+2 p x-1}{2 (1-x)^2}$$