You can recruit new Crew members from across the multiverse in a wide variety of ways.
The most obvious method to fill out your Crew quarters is by leveraging the Time Portal. A 10-pull of the Standard Time Portal pack is a good way to recruit an Uncommon Crew member (sometimes with Common Crew members to boot), and only costs Credits to use. The Premium Time Portal pack has a good chance of delivering Rare Crew, and a 10-pull guarantees at least one Super-Rare or better. Limited-time packs might give you a better chance of securing a particular Crew member, or in rare cases, even loftier guarantees (such as Legendary Crew). Check the pack info for details!
You can also recruit new personnel in your travels. Some Episodes will reward you with a Crew member representing the Faction you supported. Running a lengthy Voyage can be a great way to find extra help. Performing admirably in an Event can earn you access to the Crew associated with that Event. And if you're both very persistent and very lucky, a particularly sought-after individual might join you as a result of your victories in the Gauntlet.
Finally, procuring resources throughout the galaxy and making a good impression with various Factions can allow you to hire new Crew from the Faction Shops, as well as the Honor Hall. Bear in mind, though, that being able to pick a specific Crew member comes at a much heavier cost.