This is from the Alicante website. You could take a taxi to the Plaza del Mar or stay in Alciante and visit Benidorm which would be my preference. (You would probably need two taxis or a minibus) The playa San Juan in Alicante equals the beach in Benidorm I would say. But that is just my preference.
TRENET
Narrow-gauge train that runs along the northern coast of the province to Dénia, with stops at every interesting village, such as San Juan beach, El Campello, Benidorm, Altea, Calpe, Teulada, Gata and Dénia.
The first stop is in Plaza del Mar, at the beginning of Postiguet beach. The service between Alicante and El Campello is by tram.
FERROCARRILS DE LA GENERALITAT VALENCIANA (FGV)
Address: Avda. de Villajoyosa, 2 (at the end of Postiguet beach)
Phone number: +34 900 720 472 - Fax. +34 965 26 29 30
The train offer other services:
Trenet