I lived in Venice for three years. If you've got a modern story, curtains on gondolas are not normal--the point is to see the city. Gondolier positions are limited and you must be apprenticed and then gain one of the limited positions. Generally it's passed down through families. The gondolier knows what he's doing (and in one case, what she's doing--she made international news for breaking that gender barrier). Water taxis and other boats with engines do share canals with gondolas. They're limited--certain canals are off-limits for motorboats--but Italians aren't big rule lovers. The water taxis are the ones with the bad rep for rudely making waves, which damage the buildings. Still, a gondola would stay upright. I rowed one myself while wearing ridiculous heels (hadn't known I'd end up rowing, but I didn't want to pass up the opportunity, and it's a full body excercise) and while I could have landed myself in the canal, there was no way I would've flipped the boat or even rocked it enough to dislodge my occupants. It's traditional in Venice to take boats standing, depending on the exact circumstances of course, and the tourists who stand on the traghetti (a gondola that ferries people across the width of the Grand Canal) sure look ready to take a dive. Tourists manage to fall from dry land into canals all the time--not quite sure how they pull that off, but they're generally assumed to be amazingly clumsy. An enormous quantity of people travel through the city. Gondoliers want you to have a good time, so they're going to watch and try to keep a tourist from doing something overly dumb. The gondolier helps tourists to and from their seats, where they're quite secure.
At every blind canal corner (and the ones with mirrors), the "driver" shouts out "Oy!" to avoid a collision with someone coming from the other way. There is a hierarchy of who gets to go first depending on boat, and gondolas are at the top. The shouting back and forth that someone else mentioned is normal.
Many of the old buildings still flush all their sewage into the canals, which are regularly dredged. Only the canals that have not been dredged for some time smell. I've seen the canals all sorts of colors, and whenever the waters looked too weird, Marghera was blamed. The tide naturally flushes out the city. I know of more than one native party that ended with the entire crowd in the Grand Canal, which, being the widest, gets the clearest water exchange. An older man I knew said that when he was a child it was cleaner and the kids regularly played in the water. Generally speaking, though, natives today find the water disgusting. You can see how carefully they avoid it during acqua alta.
So, short answer: your character needs to do something very foolish or clumsy for it to be believable, but it can happen.