Can you explore the borders by car from Edinburgh easily? or better to stay in a town in the borders?
Easily. The southern edge of the Borders is only about an hour and a half out of Edinburgh. It might get a bit frustrating winding out and back in, however.
If you decide on a separate town, Jedburgh would be a good choice. On the coast, Eyemouth is attractive.
I agree with Ed and can suggest another town to stay in--Melrose. It was small, but had a lovely restaurant or two.