Collioure, France, about 45 mins by car is a lovely small town by the coast. The beach there isn't great but you could travel a few miles north to Racou, Argeles sur mer for your beach needs. Argeles is a typical tourist resort, Racou is much quieter and in my opinion nicer. If you do go to Racou or Argeles, be careful as the water floor drops very suddenly after a few feet in, if you have young kids it can be quite dangerous.
Don't know what public transport from the airport is as we've always hired a car from there but if you 'google' the airport you might get an adea.