Massimiliano Gabrieli, one of the lawyers in a pool representing about 100 passengers, said they are seeking up to 1 million euros ($1.3 million) in compensation for each survivor of the fateful cruise.
"We want this sentence to be an example, to serve as monitor so that similar accident never happen again," he said.
The civil plaintiffs also include the Costa cruise line, which is suing Schettino for causing the accident and destroying its ship.
However, many of the civil parties and Schettino's own lawyers argue that Costa and its parent company, Carnival, should be considered as criminal defendants in the trial.
'This is not a car accident'
They suggest that responsibility for the handling of the situation after the shipwreck, including any delay in the order to abandon ship, should extend beyond the captain.
But the court decided Wednesday that Costa Cruises should remain a civil party because it had suffered damage.
"This is not a car accident. I'm very disappointed that Schettino is the only defendant," Domnica Cemortan, a Moldavian dancer who was on the bridge the night of the shipwreck and is a civil plaintiff in the case, told CNN outside the court.
Evidence presented to the court is likely to include conversations between Schettino and cruise line emergency officials on the night of the accident.
The 32 deaths occurred during the abandonment of the ship rather than in its impact with the rocks.
Meanwhile, the Costa Concordia lies half-submerged on its side on the shore of Giglio Island, rusting and rotting in the waters off Tuscany's coast.
Nearly 500 workers are involved in an operation led by Titan Salvage to remove the wreck while protecting the marine environment.
Titan and Costa Cruises said last week they hope to be able to rotate the vessel to a vertical position in September. They will then be able to assess whether it can be refloated and towed away.