At the airport at Castries are only taxis and no public transport.
The road between Castries and Gros Islet is just a few minutes walk away. There are minibuses every few minutes.
Budget transport on Saint Lucia:
Minibuses depart from the capital Castries in all directions.
The minibuses have a bus number and leave when they are full.
The buses stop only at the bus stops.
Castries and Gros Islet are permanently connected from early morning until late at night. I had a bus from Castries to Gros Islet at 10 pm after the ferry arrival.
I found the most budget accommodation on AirBNB.
I can recommend staying in the capital Castries. It is centrally located and with the minibuses, you can explore the rest of the island cost-effectively.
I stayed in an AirBNB between Castries and Gros Islet. It was the most budget stay I could find with a very good price. Bus stops for minibuses were only a minute walk away to get quickly to the capital.