"An affiliate logs in after a long period of inactivity. Should I acknowledge this? And if so, what should I say?"
I would welcome them back and try to have a conversation with them to let them know I am there for them now and always and develop a friendship with them so they know I am human and care about their success too.
Then I would ask them how I can be of assistance to them providing all my details for them to contact me anytime.