Cortina Express buses run somewhat frequently from Venice Mestre train station, Venice Marco Polo airport and from Treviso airport, daily off peak season and 5 times per day in high season. Journey time from Venice to Cortina 2 hours.
There is a train to Pieve Di Cadore some 35km from Cortina that comes direct from Venice (€7.50 each way) a 6 times a day during the peak winter season and takes approximately 2h45m, and then dolomitibus runs a bus that runs roughly 5-10min after arrival that will go to go to Cortina in approx a half hour, a ticket is €3 and can be purchased upon entering. Taxis to Cortina from here are €65. Similar public transport runs in summer.