It helps if your boyfriend is aware that his mood has changed, and owns it. It's helpful if he asks for specific support that you are comfortable with and willing to give. He might need support from others, too. Also, does he need outside help like a support group? Counseling?
How does your boyfriend show his anxiety and depression? You are feeling pressure to make him feel better, but that is his job, in my opinion. I will listen to my husband, and encourage him, compliment his efforts, give him extra hugs, or let him be alone to write or exercise, sleep, etc. I might find a movie for us to watch that will let him cry. I'll make comfort food. But it's important to keep doing what you need to do for your own life, too. It's good to be independent!