Hoeffding type inequality for bounding deviation of a sequence

To simplify notations, let me assume without loss of generality that all the $X_i$'s are centered and bounded by 1 in absolute value. For sure, one trivial upper bound to your quantity is $$ \mathsf P\left(\bigcap_{n\le N} (S_n\ge nt)\right)\le \mathsf P\left(S_N\ge Nt\right)\le e^{-t^2N/2}. $$ In fact, you cannot hope to do much better than this in general. Indeed, you can always lower bound $$ \mathsf P\left(\bigcap_{n\le N} (S_n\ge nt)\right)\ge \mathsf P\left(\bigcap_{n\le N} (X_n\ge t)\right)=\mathsf P\left(X_1\ge t\right)^N. $$ And we can make the right-hand side arbitrarily close to the upper bound, at least up to universal constants (for example, take a standard Gaussian and truncate it to stay in $[-1,1]$).