From the viewpoint of the Wightman axioms, the separability assumption on the Hilbert space can be actually derived from a few of the other axioms if you adopt a formulation using $n$-point functions.
The reasoning goes as follows. A (say, scalar) quantum field theory on $\mathbb{R}^d$ can be thought of as being specified by a sequence of $n$-point distributions $\omega_n\in\mathscr{D}'(\mathbb{R}^{nd})$, $n=1,2,3,\ldots$ satisfying a condition of positivity which will be specified shortly (the other Wightman axioms are not relevant for the purpose of establishing separability of the "vacuum" Hilbert space).
The point is that this sequence of distributions can be thought of as a linear functional $\omega$ on the algebra $\mathfrak{F}=\mathbb{C}\oplus\left(\bigoplus^{\infty}_{n=1}\mathscr{D}(\mathbb{R}^{nd})\right)$ - an element $f=(f_0,f_1,f_2,\ldots)\in\mathfrak{F}$ is a sequence such that $f_0\in\mathbb{C}$, $f_n\in\mathscr{D}(\mathbb{R}^{nd})$ are zero for all but finitely many $n\in\{0,1,2,\ldots\}$. The algebraic operations on $\mathfrak{F}$ are defined in the following way: if $f,g\in\mathfrak{F}$ and $\alpha\in\mathbb{C}$, then we write:
- Sum: $f+g=(f_0+g_0,f_1+g_1,f_2+g_2,\ldots)$;
- Scalar multiplication: $\alpha f=(\alpha f_0,\alpha f_1,\alpha f_2,\ldots)$;
- Product: $fg=((fg)_0,(fg)_1,(fg)_2,\ldots)$, where $(fg)_0=f_0g_0$ and $(fg)_n(x_1,\ldots,x_n)=\sum_{i+j=n}f_i(x_1,\ldots,x_i)g_j(x_{i+1},\ldots,x_n)$;
- Involution: $f^*=(f_0^*,f_1^*,f_2^*,\ldots)$, where $f_0^*=\overline{f_0}$ and $f^*_n(x_1,\ldots,x_n)=\overline{f_n(x_n,\ldots,x_1)}$.
The product is just the tensor product of test functions, whereas the involution (a sort of noncommutative analogue of complex conjugation) makes $\mathfrak{F}$ into a unital $*$-algebra. This *-algebra inherits a (locally convex) topology from the test function spaces $\mathscr{D}(\mathbb{R}^{nd})$ which makes it a so-called nuclear *-algebra. The sequence $\omega=(\omega_0,\omega_1,\omega_2,\ldots)$, where $\omega_0=1$, becomes a (continuous) linear functional on $\mathfrak{F}$ if we set $\omega(f)=f_0+\sum_{n=1}^\infty\omega_n(f_n)$ (the sum is always finite by our definition of $\mathfrak{F}$ above). The *-algebra $\mathfrak{F}$ is sometimes called a Borchers-Uhlmann algebra.
Now we can state our condition of positivity on $\omega$: for all $f\in\mathfrak{F}$, we must have $\omega(f^*f)\geq 0$. In other words, $\omega$ is a(n algebraic) state on the *-algebra $\mathfrak{F}$.
At this point, we can invoke a result due to K. Maurin ("Mathematical Structure of Wightman Formulation of Quantum Field Theory", Bull. Acad. Polon. Sci. 9 (1963) 115-119), which essentially tells us that the nuclearity of $\mathfrak{F}$ and the continuity of $\omega$ entail that the Hilbert space obtained by the Wightman(-GNS) reconstruction theorem is separable. Notice that the actual construction of the Hilbert space and the "vacuum" vector need only positivity to work. The other axioms (covariance, causality) are needed to obtain the unitary representation of the Poincaré group, spectrum condition, and so on. Therefore, Maurin's argument is stronger (and simpler) than the one found in Streater-Wightman's book.
The argument can be extended to fields of any spin, provided we define the tensor product of test sections of vector bundles in the appropriate way. I don't know of an analogue of this argument for fields where the requirement of positivity is not satisfied (e.g. electromagnetic fields in a covariant gauge). However, for free fields the Wightman reconstruction theorem is just the construction of the vacuum Fock space from the one-particle space, which always yields a separable Hilbert space. In general, one may think of such Hilbert spaces as "sectors", as argued in David Bar Moshe's answer.
Finally, one must recall that there are other Hilbert spaces which are interesting for quantum field theory and are non-separable. Of course, these cannot be obtained by the Wightman reconstruction theorem alone. One such example is provided by all coherent states of a free field. One can use Wightman reconstruction with a single coherent state, but the resulting Hilbert space cannot contain all coherent states. Such spaces also appear indirectly in the Bloch-Nordsieck approximation in QED, used to deal with infrared problems. They are a particular case of the continuous tensor product of Hilbert spaces mentioned in David Bar Moshe's answer, which also discusses other examples of interest.
