Check on line for Mass times at St Marks. Then go to the side door prior to Mass time, and you will be admitted. A few years ago, we attended Mass this way, and afterwards, had the entire main floor to ourselves as the "tourists" had not been allowed in. We were also up on the main altar, an area not accesible to the normal visitor.
It was worth getting there early!
The following is from the Basillica's web site:
The holy masses celebrated in the basilica are free of charge and you do not have to make a reservation as for the visits.
The entrance is from the Porta dei Fiori (north fašade) in the Piazzetta dei Leoncini.
07:00 a.m.; 08:00 a.m.; 09:00 a.m.(Sung Mass of the Charter of the Canons preceded by the celebration of the praises); 10:00 a.m. (in baptistery); 11:00 a.m.; 12:00 (suspended in the months of July and August); 06:45 p.m. (preceded by the celebration of Vespers and the rosary).