Hints: Let $S_n=\sum\limits_{k\leqslant n}X_k$.
(1.) The event $[n^{-1/p}S_n\ \text{bounded}]$ is included in the event $[n^{-1/p}X_n\ \text{bounded}]$.
(2.) For each $x\gt0$, the series $\sum\limits_n\mathrm P(|X_1|\gt xn^{1/p})$ converges if and only if $X_1$ is in $L^p$.
(3.) By Borel-Cantelli, if the series $\sum\limits_n\mathrm P(|X_n|\gt xn^{1/p})$ diverges, then $\limsup\limits_nn^{-1/p}|X_n|\geqslant x$ almost surely.
(4.) Assume that $X_1$ is not in $L^p$, then use the contrapositive of (2.), then (3.), and finally (1.) to conclude that $n^{-1/p}S_n$ is not almost surely bounded.