My favorite is Hostel One Sants. I've stayed there twice and both times were amazing. They do a free dinner which is really nice and allows you to connect with the other guests. That is where I (was supposed to) start my camino last year.
The Camino House B&B in the Montjuic neighborhood (quiet) has private rooms and dorms. Starts at 30 Euros/night. http://www.caminobnb.com/ If you want to party, you're on your own. I really like the Hostal Girona https://www.hostalgirona.com/ in the Eixample neighborhood. Both are bookable directly or on www.booking.com. Have a great trip. I'm in Barcelona now!