Why does sea level correspond to boundary between oceanic and continental crust?
Sea-level only sort-of corresponds to the oceanic–continental crust boundary.

In depth, they don't correspond at all:



It's the same story at an active margin: the plate boundary at a subduction zone is buried several kilometres beneath a wedge of sediment:



What about spatially? As you can see from this map, they don't correspond all that well spatially either: they are commonly 100s of kilometres apart, sometimes more.



The difference, shaded grey in the map, is the continental shelf. It is sometimes emergent (e.g. at the end of the Permian, or during ice ages, when global sea-levels are low). It is sometimes, as now, flooded. The vertical change of ca. 100–250 m means a lateral change of 100s of km, because the shelf is almost flat.

OK, enough nit-picking, roughly speaking, why are the oceans on oceanic crust?

Continental crust is relatively thick and low density (ca. 2700 kg/m$^3$), compared to oceanic crust (25–70 km and ca. 2900 kg/m$^3$). So it 'floats' higher on the thin, dense mantle (7–10 km and ca. 3300 kg/m$^3$). When a new rift forms, it creates a depression and space for new oceanic-style crust. This is happening right now in the Gulf of California, the Baikal Rift Zone, and in the extensive East African Rift system. All of these places have large lakes and/or are extensively below sea-level. It's a divergent 'tear' in the continental crust, so intuitively we might expect a depression. As it fills with thin oceanic crust, the depression is maintained and eventually fills with ocean water.

Why is ocean crust so thin? I don't have a lot of expertise here, but I think it's determined by the relative rates of melting and cooling, and the heat balance at the ridge and subduction zones. There's a nice answer to this question on Reddit, complete with links. As insinuated in that thread, the thermo- and chemo-dynamics of the spreading centres and subduction zones are intimately related to the presence of the ocean, so together they make a complex system (indeed, the entire planet is a complex system), as @stali points out in his comments and answer.