I planted begonias as a border and to provide color. It took a small hit during the last frost of the year. I thought they would take hold and come out of it. They look puny. I see many places that their begonias are big and beatiful in full sun.
At first I thought the sun was too much. What should I do? Replace them, if so, with what type of border flower?

It could be the type of begonias you are using. There are so many. Why not try native plants. You do not give any clue as to where you live, but if you live in the DFW area, "Plants of the Metroplex" by John Howard Garrett is an excellent reference tool!