Yes, a basic phone CAN be activated on a line with a plan "designed" for a smartphone. The plan would simply change, if needed. I suppose you could say there are "plans designed for smartphones", but for the most part plans are interchangeable between smartphones an basic phones. Family "plans" have both smartphones and basic phones on them, so THOSE plans are not necessarily "designed for smartphones". The access fee may change if you change from a smartphone to a basic phone, but the plan stays the same.
For older and/or prepaid plans there will be a data plan associated with the smartphone which may or may not have to remain in place if switching to a basic phone, but that would simply be a cost savings if the data plan could be removed.