Yes, you should worry that the dog might hurt or even kill the cat.
I wouldn't say, "NO!" to him when he growls or otherwise is trying to communicate with the cat that he doesn't want him near the crate. If you shut off his ability to communicate, he'll stop doing that and suddenly and swiftly do what he's trying to avoid doing (with that communication).....lash out and bite.
What I would suggest is, when you're right there to supervise and the cat is near by...(keep him at a distance though, that is tolerable for the dog...set up situations purposely) feed the dog tiny, scrumptious treats and lots of praise. So, in other words, he learns that the cat's presence is paired with really good things. Do this for a minute or two several times a day. When you can't be training or explicitly supervising, I recommend you keep the cat in a safe place. Save the treats and hand feeding even, of some of his dinner for these training sessions. Have someone else in your family keep the cat back if you can to help you practice this safely. When you're done with a training session, treats stop. The most fun and the best treats should be reserved for associating with the cat's presence.
You might also begin feeding the dog in various locations rather than in the crate, as well as sleeping in a few different places, making those other places really special. Lower the value of the crate. But again, be very careful that your cat doesn't go too close....really watch or remove the cat from the room while he eats if you can't be supervising. Watch for that body language; stiffening, ears back, stares, freezing up....growls, etc. Those are warning signs and should be respected. You'll need to remove the cat to ensure nothing bad happens.
Those little training exercises might help a little. But don't put all your confidence in that training. There's still instinct to defend one's resources.
Employ some NILIF (Google it) with your dog. Remind him that YOU control his resources. Have him earn the good stuff. He needs to feel secure that you will keep the cat from threatening his resources, but that the things he likes are contingent on his behavior. You can even try having him sit/wait for a second before getting into his crate, if he loves his crate.
If you don't feel comfortable about the dog, crate, cat issue, you might want to remove the crate from a room the cat goes in. And only use it when you must. Of course, the dog can just as well transfer that resource guarding to another thing....the couch or bed, whatever. But anyhow, don't pair a scolding or "no" with the presence of the cat. That can make him even worse. So practice some deferential training with him, if you're not already doing so.
And remember, cats are considered prey to a lot of dogs, even if he's been fine for a while, as a pup. Things can change, especially as the dog gets more settled into his new environment. And if that's Corky in your siggy, (he's very cute btw) this type of dog can be particularly prey driven and it's not uncommon for them to not dig other animals in an up close and personal way.