Not just the Erie Canal but the C&O Canal before that, and various canals in England like the Trent and Mersey Canal, all long before electricity was commonly available.

Canals work, first, because water runs downhill. Every canal would have a beginning at a higher water source- in the case of the Erie, Lake Erie, which would flow downhill to Albany. There could also commonly be "feeder "canals, streams and other water sources would be channelized and fed into the main canal. Then there would be a series of locks, essentially double doors that could be opened and shut: you can see a time-lapse example of the Erie here. It's essentially the energy of that water flowing downhill that is being used to lift or lower boats. The second reason canals work is that a slow-moving boat has amazingly little friction as it moves through still water- that's why a couple of mules could be the entire motive force for a canal boat carrying 30 tons. None of that requires electricity.

But electricity would be useful for a canal. The gates had large levers that were pushed to open and close them, as you can see here. The amount of drop in the height of water was limited by the height of those lock gates. Big gates were harder to swing than little gates, of course, so those manually-operated gates had to be limited in size. The Erie Canal originally needed five locks at Lockport to descend the Niagara Escarpment, each capable of lifting a boat about 8 feet. There were improvements in the canal through the 19th c., until it was re-named the New York Barge Canal, but those five locks were replaced in 1902 by two locks, each lifting about 24 feet, that are presently in operation. Those big lock gates were impossible to operate by a man pushing on a long lever- they were too heavy. I am not sure if they were always operated by electricity, or whether there was a brief period in which they were moved by a steam engine. But certainly they are operated by electricity now. The advantage of using two big locks instead of five small ones is time: it takes a good while to swing one set of the lock gates shut, drain the water, open the next set. There are various other electrical aspects of the canal system now- warning bells, and lights- but it's the lock gates which are unavoidably operated by electricity.

There is a last step to a canal- loading and unloading the boats! In the first days of the Erie Canal, men with shovels would do that. At the end of the 19th c., conveyors and other machinery would be used to do the job considerably faster.

McIlwraith, T. F. (1976). Freight Capacity and Utilization of the Erie and Great Lakes Canals before 1850. The Journal of Economic History, 36(4), 852–877. http://www.jstor.org/stable/2119243