There are multiple trails in the CT, and they aren't all equally popular. The main trails closer to the ocean between the five towns are the most popular. Between Riomaggiore and Manarola and Manarola to Corniglia are still closed I think. Between Corniglia and Vernazza and Vernazza and Monterosso are the most busy. If you want to hike them, get out there first thing in the morning or wait until later in the day.
Meanwhile, make alternate plans. The alternate hike between Manarola and Corniglia - a steep hike up, up, up to the little town of Volastra and then down to Corniglia is not nearly as popular - it's a much steeper climb, though it's easy enough if you take your time and pace yourself. It's not without hikers - you'll see plenty just not as many as the main trails. As a bonus, the views down from this hike are truly breathtaking, maybe the best of any of the hikes I've done in the area.
You can also hike between Levanto, (next town north of Monterosso) back to Monterosso. While this hike isn't as nice as the others, it has definitely highlights, and it's not nearly as crowded. The view down from the point above Monterosso gives you a view of all five towns at once. Great spot.
The Italian Riviera has other beautiful places besides the Cinque Terre. That's just the most popular part of it. About an hour north of the Cinque Terre is the Portofino peninsula with beautiful towns like Rapallo, Santa Margherita Ligure, and Camogli with some beautiful hikes that are much, much less busy than down in the Cinque Terre.
In the Cinque Terre, don't plan on being in the villages proper in the daytime when they are so busy - plan to be elsewhere. They are much nicer in the evening when most of the day trippers have left.