Here is how to solve the basic task in $O(n)$, at the expense of possibly lots of preprocessing. (This expands a comment I made.)
Example. Suppose that $n$ is 3, and the given sets are {1,2}, {2,3}. Represent the boolean function $f(x_1,x_2,x_3)=(x_1\land x_2)\lor(x_2\land x_3)$ as a BDD:
$$f(x_1,x_2,x_3)=x_1?(x_2?1:0):(x_2?(x_3?1:0):0)$$
(Here $x?a:b$ means 'if $x$ then $a$ else $b$'.)
Clearly, evaluating $f$ written in this form takes time linear in $n$, which is possibly much smaller than the size of the input. The problem, of course, is that going from DNF to BDD may take an exponential amount of time (and space). But this is just preprocessing.
Suppose the query is the set {1}. We evaluate $f(0,1,1)$ to get 1, so we conclude that one of the given sets is disjoint from the query.
Suppose the query is the set {2}. We evaluate $f(1,0,1)$ to get 0, so we conclude that none of the given sets is disjoint from the query.
The general case. Given sets $S_1,\ldots,S_m\subseteq[n]$, represent the function
$$f(x_1,\ldots,x_n)=\bigvee_{i=1}^m\bigwedge_{j\in S_i}x_j$$
as a (RO)BDD. To answer a query $X$ evaluate $f(x_1,\ldots,x_n)$ with $x_j=[j\notin X]$ for $1\le j\le n$.
Comments. One way to view this construction is as a systematic way to precompute all answers. ‘Precompute all answers’ sounds dissatisfying, but I think the keyword should be ‘systematic’: it's not a priori obvious how to do it for this problem, given the comments on the question.
Another thing to note is that there are many negations going on here, which some people say are obvious, but they tend to make me dizzy. Probably the only reason I saw this construction quickly is that minutes before reading the question I was looking at the Monotone Duality problem, which is closely related to the question. Monotone Duality comes in many guises. One of them is Hitting-Sets: Given a family of sets, compute the family of all the hitting sets. This is clearly related to the question posed here, in which we are asked whether a given set is not a hitting set. (It's not the same though.) Another guise is Monotone-DNF-Dual: Given a monotone $f$ in DNF, find a $g$ in DNF such that $\lnot f(x_1,\ldots,x_m)=g(\lnot x_1,\ldots,\lnot x_m)$. I thought of the construction above because I knew that this is an equivalent formulation. So, perhaps a reference to a survey of the Monotone Duality problem would serve as a suitable reference:
What does this construction say about lower bounds? Not much. But it does say that if you find a solution that answers a query in $O(h(m,n))$ time after $O(g(m,n))$ time preprocessing, then there exists a representation for monotone boolean functions that takes $O(g(m,n))$ space and allows evaluation in $O(h(m,n))$ time. For example, BDDs have exponential $g$ and $h(m,n)=n$. The reference for ‘exponential $g$’ is
PS: I would be grateful for any comments on how to improve this answer.