There was an article not too long ago about IBM making the smallest world's magnetic computer memory bit - twelve atoms. Smaller than that, and quantum effects messed up its ability to store data. Of course, it required a tunneling microscope, and heating/transfer rates mean its probably not practical.
We could probably make electronic computers without quantum mechanics. We could probably even step past vacuum tubes to transistors, even on to microchips - but probably not with the same sort of Moore's law doubling. That is just my guess, however. It may be quite possible that all radio wave transmitters - even the spark gap transmitter (the tube's predecessor) relies on a quantum effect, just never described as such.
Of course, a world without microchips is an interesting one indeed, even without the retro Victorian veneer. Microchips are what made unmanned satellites smaller and cheaper, and thus a better alternative to manned space missions - computers killed the need for space stations. Advanced computers (and modeling made with them) and miniaturized components shrunk the size of nuclear weapons - making ICBMs possible, as wells as most guided weapons. No microchips, and modern air raids would look like something out of WWII, just with faster aircraft.