Dally's biggest fear is that something will happen to Johnny.
Dally is "of the breed that (can) take anything, because he (is) hard and tough, and when he (isn't), he could turn hard and tough". He grew up in the streets and was in jail the first time at the age of twelve; he is a hardened character who has learned that by not letting himself care about anyone or anything, he will not get hurt. Dally has a soft spot for Johnny, however. Recognizing that the younger boy has had a life every bit as hard as his own, Dally is uncharacteristically touched by the sensitivity Johnny still retains. Dally cares about what happens to Johnny.
When Johnny tells Dally that he wants to turn himself in, Dally is concerned. He tells Johnny, "I just don't want you to get hurt. You don't know what a few months in jail can do to you...you get hardened in jail. I don't want that to happen to you. Like it happened to me".
Dally ordinarily doesn't "give a Yankee dime about anyone but himself...he (is) cold and hard and mean". He has put down his guard only where Johnny is concerned, and his greatest fear is that something bad might happen to the only person he cares about in the world (Chapter 6).
he does not want johnny to die
I'm not sure if this is his biggest fear but it's definitely a fear that he's got: He oesn't want Johnny to get hardened in jail like he did.