All players start with a total of 80 Crew Slots. If you start to run out, you can purchase more by tapping the + sign at the bottom-right of your Crew Quarters.
Crew Slots are purchased in batches of 5, and can be purchased up to 63 times. The cost increases with each batch purchased, as follows:
Batch 1 (5 total): 250k Credits
Batch 2 (10 total): 250k Credits
Batch 3 (15 total): 350 Dilithium
Batch 4 (20 total): 700 Dilithium
Batch 5 (25 total): 1050 Dilithium
Batch 6 (30 total): 1750 Dilithium
Batch 7 (35 total): 2450 Dilithium
Batch 8 (40 total): 3000 Dilithium
Batch 9 (45 total) - Batch 63 (315 total): 3000 Dilithium
You can also earn Crew Slots by advancing your VIP level. The total number of Crew Slots available from all sources is currently 515. If you want to collect every Crew in the game, be sure to strive for Immortalization so that you can place unneeded Crew in Cryostasis.