The property in question is a special case (with the probabilities of all the $k$ outcomes equal to one another) of the known $NA$ (negative association) property of the multinomial distribution; see e.g. this sentence in the bottom paragraph on page 5:
$NA$ property of multinomial distributions can be seen from Condition $N$, since it is the conditional distribution of independent Poisson random variables given their sum.
Indeed, for each $i\in[k]:=\{1,\dots,k\}$, let $N_i$ denote the number of times the letter $i$ appears in the random word of length $n$. Then $(N_1,\dots,N_k)$ has the $k$-nomial distribution with parameters $n,1/k,\dots,1/k$, and the inequality in question can be written as $$P(N_1\ge p,\dots,N_s\ge p)\le P(N_1\ge p,\dots,N_{s-1}\ge p)P(N_s\ge p)$$ or, equivalently, as $$Cov\,\big(f(N_1,\dots,N_{s-1}),g(N_s,\dots,N_k)\big)\le0,$$ where $f$ and $g$ are the functions (nondecreasing in each argument) given by formulas $$f(n_1,\dots,n_{s-1}):=1_{n_1\ge p,\dots,n_{s-1}\ge p}$$ and $$g(n_s,\dots,n_k):=1_{n_s\ge p}\,;$$ cf. Definition 2.1 of $NA$ in the linked paper.