How far apart should nails be when installing baseboards?
We usually tack the baseboard in place with one nail at each side, and then find the studs with a stud sensor, marking on the baseboard as we go. 

Alternately, I would think that every 12 inches would be sufficient. We also caulk the top of the baseboards with paintable latex caulk (which helps with dusting later), and that helps hold the baseboard to the wall too. 

With shoe moulding (which I seriously recommend, especially on 2nd stories and pier on beam structures), it doesn't matter since the nails go into the baseboard first.