Let $(\Omega,\mathcal F,P)$ be a probability space, and consider a random variable $X: (\Omega, \mathcal F)\longrightarrow (S,\mathcal S)$, where $S$ is a separable real Banach space. It is well-known that $X$ induces a probability measure $\mu$ on $(S,\mathcal S)$.
In the case $S = C(T)$ or $S = L^p(T)$, with $T\subset\mathbb R$, the probability measure $\mu$ is uniquely characterized by its finite dimensional marginal distributions $(\pi_J)_*\mu$ with $J\subseteq T$ finite. Here $(\pi_J)_*\mu := \mu\circ\pi^{-1}_J$, and $\pi_J^{-1}$ denotes the preimage of the coordinate projection $\pi_J:S\longrightarrow\{\text{all functions $J\longrightarrow\mathbb R$}\}$ with $\pi_J(f) = (f(j))_{j\in J}$.
In a general separable Banach space, we don't have coordinate projections. Does the concept of marginal distribution exist nevertheless? If yes, how is it defined?
There is indeed a very good concept of finite dimensional marginal distributions of $\mu$ when $S$ is a general separable Banach space, or more generally when $\mu$ is a Radon probability and $S$ is a real topological vector space that is separated by its dual. This is tied to the notion of "cylindrical probability" on $S$.
Cylindrical probabilities
Denote by $\mathscr F$ the set of all closed linear subspaces of $S$ with finite codimension in $S$ and for every $E\in\mathscr F$, denote by $\pi_E$ the quotient map from $S$ to $S/E$. The set $\mathscr F$ is upward directed for the order given by $\supseteq$. So for every $E,F\in\mathscr F$ such that $E\supseteq F$, if you denote by $\pi_{FE}$ the unique linear map from $S/E$ to $S/F$ satisfying $\pi_F=\pi_{FE}\circ\pi_E$, the finite dimensional quotients $S/E$ together with the maps $\pi_{FE}$ form a projective system of vector spaces.
A cylindrical probability on $S$ is a projective system of probabilities on the projective system of vector spaces given by the quotients $S/E$ and the maps $\pi_{FE}$, i.e. a family $(\lambda_E)_{E\in\mathscr F}$ where each $\lambda_E$ is a Radon probability on $S/E$ satisfying $\lambda_F=(\pi_{FE})_\ast(\lambda_E)$ for every $E,F\in\mathscr F$ such that $E\supseteq F$.
Link with Radon probabilities
Now, when $\mu$ is a Radon probability on $S$, you can consider the family $((\pi_E)_\ast\mu)_{E\in\mathscr F}$ of the "finite dimensional marginal distributions" of $\mu$. This family is a cylindrical probability on $S$ and it satisfies a very particular "tightness" condition : for every $\varepsilon>0$, there is a compact subset $K$ of $S$ such that
$$\forall E\in\mathscr F,\qquad\Bigl((\pi_E)_\ast\mu\Bigr)\Bigl((S/E)\setminus\pi_E(K)\Bigr)\leq\varepsilon.$$
It can be shown that conversely, a cylindrical probability on $S$ that satisfies this tightness condition is the family of finite dimensional marginals of one and only one Radon probability on $S$. This can be seen as a consequence of a theorem of Prokhorov concerning the existence and uniqueness of projective limits of bounded Radon measures. So in particular, the finite dimensional marginals of a Radon probability $\mu$ completely determine $\mu$.
Some references
You can find out about Prokhorov's theorem in Laurent Schwartz' book Radon measures on arbitrary topological spaces and cylindrical measures (Zbl 0298.28001), part I, chap. I, §10, thm. 22. It should also be present in Prokhorov's original paper (but I couldn't pinpoint the exact whereabouts of the relevant theorem) and you can also get it from Bourbaki's Integration, chap. IX, §4, n°2, thm. 1.
If you want to learn more about cylindrical probabilities, you can check out the part II of L. Schwartz' book and the chap. IX, §6 of Bourbaki's Integration. Aside from the "projective system" point of view on cylindrical probabilities, there is an "additive set function" point of view which you can get from Gel'fand & Vilenkin's book Generalized functions. Vol. 4: Applications of harmonic analysis (Zbl 0136.11201), chap. IV.