With $\{ξ_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $ρ(i):=\mathbb{E}[ ξ_0ξ_i]$ and $\{σ(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum $\sum_{i=1}^\ell σ(i) ξ_i$, $\ell\ge 1$. For summable correlations $ρ$, we show that the persistence exponent is universal. On the contrary, for non-summable $ρ$, even for polynomial weight functions $σ(i)\sim i^p$ the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter $H$) and on the polynomial rate $p$ of $σ$. In this case, we show existence of the persistence exponent $θ(H,p)$ and study its properties as a function of $(p,H)$. During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations -- e.g.\ a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero -- that might be of independent interest.