 Solution

Every linear combination of $X_{t_1},\dots ,X_{t_n}$ is a deterministic constant plus one stochastic integral of a deterministic $L^2$ function against $B$. Part a makes every such combination Gaussian. By the linear-combination characterization of a multivariate normal distribution, $(X_{t_1},\dots ,X_{t_n})$ is jointly Gaussian, so $X$ is a Gaussian process.