Are there more connected or disconnected graphs on $n$ vertices?

Connectedness wins, since the complement of any disconnected graph is connected.

EDIT: Perhaps you'd like a proof of this. Let G be a disconnected graph, G' its complement. If v and u are in different components of G, then certainly they're connected by an edge in G'. And if they're in the same component of G, then there's some w in another component (since G was disconnected), so v-w-u is a path in G'.

For large $n$, not only are the vast majority of graphs on $n$ vertices connected, the vast majority have diameter 2. That is, any two vertices have a neighbor in common. (The standard reference for properties of most graphs on $n$ vertices, for large $n$, is the book "Random Graphs" by Bela Bollobas.)

Connectedness wins by a knockout: the proportion of disconnected graphs is about $n2^{-n+1}$. See Flajolet, Sedgewick "Analytic Combinatorics", p. 138.