It has recently been shown that reinforcement learning can be used to train generators capable of producing high-quality game levels, with quality defined in terms of some user-specified heuristic. To ensure that these generators' output is sufficiently diverse (that is, not amounting to the reproduction of a single optimal level configuration), the generation process is constrained such that the initial seed results in some variance in the generator's output. However, this results in a loss of control over the generated content for the human user. We propose to train generators capable of producing controllably diverse output, by making them 'goal-aware.' To this end, we add conditional inputs representing how close a generator is to some heuristic, and also modify the reward mechanism to incorporate that value. Testing on multiple domains, we show that the resulting level generators are capable of exploring the space of possible levels in a targeted, controllable manner, producing levels of comparable quality as their goal-unaware counterparts, that are diverse along designer-specified dimensions.