How did cyclists get over curbs before BMX era?
People were doing tricks on bicycles almost as soon as they were invented. While I couldn't find references about curb hopping in particular, I'm pretty sure that these tricksters from the late 1800s wouldn't have had a problem pulling it off.  

Cyclocross has roots that go almost as far back. The first organized cyclocross races were around the end of the 19th century and the first French National Championship was in 1902.

I would guess that prior to BMX most people got over curbs the same way that most people get over curbs today. They slowed down a bit, lifted the front wheel over the curb, and leaned forward to get the weight off the rear wheel. Keep in mind, most cycling commuters can't pull off a decent bunny hop and many commuter bikes aren't designed to do so.