From St Javier airport a taxi is 30-35 euros. A bus runs from St Javier to Balsicas I think 3 times a day but that means you have a 2-3 mile walk each end and it is unlikely the bus would be at a convenient time for your flight.
I think there is a hourly bus service from Alicante airport during the day and early evening to Murcia and then you could get a bus or train from Murcia to Balsicas but I think there are only 3 or 4 a day of each.
I don't know the cost of a taxi from Alicante.