I disagree with staying in Salerno, it is nothing like Positano. However, Amalfi town, Capri and Ravello are in my mind good alternative (yes, even with getting in and out of Ravello because of those fantastic out of this world views).
But honestly, stay in Positano. This is the town that you chose, I am assuming for good reasons: it is postcard perfect, it is small, it is probably a place you have though about for a while.
I have been in Positano four times. Last time I chose Alcione Residence because of how spacious the apartment was and the view on the sea. It was also in my budget since it was off season.
Walking in, I could hear the waves crashing and the balcony had the iconic view of Positano. It had a bedroom, a small kitchen, a washing machine, a living room with a dinning room table. Breakfast is included and the management brings it to you in the morning. Next door to the Alcione Residence was Le Sireneuse. Shortly after checking in, we went for a late lunch at Le Sireneuse. The views were the same as our apartment but the grounds were beautiful and the service impeccable. The sun came and once again, it was as if it was a nice summer day without the crowds because we were there in October.