Strained tomatoes, called "passato" in Italian, are traditionally used in making meat sauces for pasta.
Place tomatoes in a large stockpot, working in batches, if needed. Bring to a boil over medium-high heat, stirring occasionally. Add salt, and reduce heat to medium; simmer 15 minutes, uncovered. Remove from heat; let cool slightly.
Using a ladle or large spoon, skim off the released juices that have accumulated in the pot; set aside in a large bowl (depending on the size of the tomatoes, you should have 4 to 7 cups liquid). Pass tomatoes through a food mill on the finest holes into a separate large bowl.
Place a large basil leaf in the bottom of each pint or quart jar; fill with tomatoes. Clean and dry top rim of each jar with a kitchen towel, and place lids and rings on jars; twist tightly to seal.
Fill a large canning pot lined with a wire rack with water, and bring to a boil over high heat. (If you dont have a canning pot, you can fit a large stockpot with a wire rack or with a layer of extra ring lids to keep the jars from sitting directly over the heat. The stockpot should be deep enough that the jars will be covered with water when they are added.)
When the water has come to a full boil, carefully place the jars in the water with tongs or a jar clamp. Boil pint jars 30 minutes and quart jars 45 minutes.
Carefully transfer jars to a baking sheet; let cool. When completely cool, check the lids to make sure they are sealed by pressing down on the middle. If the lid pops back, it is not sealed; refrigerate unsealed jars immediately, and use within 2 weeks. Sealed jars can be stored in a cool, dark place 2 years or longer.
The accumulated juice can be saved in separate jars; it is rich in flavor and can be used instead of vegetable stock.