The cheapest way to get to Livorno from Pisa Airport, and vice versa, is by using local transport:
By train - Pisa airport is linked to the Pisa Centrale train station by the continuous "Pisa Mover" train connection which costs 2.70 euros.
From Pisa Centrale there are regular direct trains to Livorno (€2.50 each way on Regional trains). Please see the Trenitalia website for the full timetable.
When all else fails, a taxi from the airport to Livorno costs around 50 euros, depending on time of day and amount of luggage.
Alternatively, you might want to book a private transfer from Pisa Airport to Livorno