Well, it's quite clear that Steph is going to go out and puch criminals with or without Babs' help. Letting her get killed due to bad intel isn't exactly Oracle's style.
And I don't doubt that an alarm would go off in Dick's new base the second someone set foot in the cave, even if they had left it empty. Considering that Babs had time to have Steph's Batsuit made, she probably had time to tell Dick "Hey, I want to use that Batcave for something. Could you leave the computers there?" And considering how well Babs kep her own security running, I don't think keeping the cave running would be too much trouble for her.