Opacity is an important information-flow security property which characterizes the plausible deniability of certain 'secret behaviors' in dynamical systems. In this paper, we study the problem of synthesizing controllers enforcing a notion of opacity over discrete-time control systems with continuous state sets. In this paper, we develop an abstraction-based approach to tackle the controller synthesis problem. Specifically, we adopt a notion of approximate opacity which is suitable for continuous-space control systems. We propose a notion of approximate initial-state opacity preserving alternating simulation relation which characterizes the closeness between two systems in terms of opacity preservation. We show that, based on this new notion of system relation, one can synthesize an opacity-enforcing controller for the abstract system which is finite and then refine it back to enforce opacity over the original control system. Finally, we present a method for constructing opacity-preserving finite abstractions for discrete-time control systems under some stability properties. Our results are illustrated on a two-room temperature control problem.