We consider random subshifts of finite type in the following sense. For natural numbers $n$ and $d$, a real number $0 < p < 1$, and a finite color set $A$, we define a random subset of the set of all colorings of $\{1,2, …, n\}^d$ by including each element with probability $p$, and excluding it with probability $(1-p)$. Extending results of McGoff and Pavlov, we prove there exists $p_0 > 0$ such that for $p < p_0$, with probability tending to 1 as $n$ tends to infinity, the associated random subshift will contain only finitely many elements. In the one-dimensional case, we obtain the best possible $p_0$.