Basically, you make a copy of the tree (or bush or vine, whatever)

What you do is you plant any old seed for that fruit until you get a seedling. Then you take a living branch from the tree that makes the seedless fruit and you replace the top of the seedling with it. This is called grafting. Basically, the seedling will repair itself with that new branch and continue growing from that new branch until it becomes a fruit bearing tree, making new seedless fruit. The beginning stages of the graft are the most dangerous because the seedling basically has an open wound and a very weak connection, but once that heals over its just like caring for a normal tree.

This is exactly what we have to do for apples because when apple trees reproduce, their offspring is very different from its parent, and very unpredictable. It makes it hard to get good apples from seeds, so we use grafts to essentially cline existing trees that we know make good apples. This also means that most fruit you eat is genetically identical (ie all honeycrisp apples are the same, all strawberries are the same)

This does lead certain varieties of fruit to be prone to disease. For example, the dominant banana in supermarkets in the 50s was the Gros Michel, and it tasted like the banana flavoring we have today (because the flavor was based on that variety of banana) but most of the trees died out due to a fungus that specifically attacked that tree. Bananas we have today are a different variety called the cavendish banana. They could easily fall victim to a similar fate, because like other fruits, they are all genetically identical to each other. We do have other varieties of banana produced in smaller quantities that are prepared to take over should the cavendish die out.