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Target speech extraction is a technique to extract the target speaker's voice from mixture signals using a pre-recorded enrollment utterance that characterize the voice characteristics of the target speaker. One major difficulty of target speech extraction lies in handling variability in ``intra-speaker'' characteristics, i.e., characteristics mismatch between target speech and an enrollment utterance. While most conventional approaches focus on improving {\it average performance} given a set of enrollment utterances, here we propose to guarantee the {\it worst performance}, which we believe is of great practical importance. In this work, we propose an evaluation metric called worst-enrollment source-to-distortion ratio (SDR) to quantitatively measure the robustness towards enrollment variations. We also introduce a novel training scheme that aims at directly optimizing the worst-case performance by focusing on training with difficult enrollment cases where extraction does not perform well. In addition, we investigate the effectiveness of auxiliary speaker identification loss (SI-loss) as another way to improve robustness over enrollments. Experimental validation reveals the effectiveness of both worst-enrollment target training and SI-loss training to improve robustness against enrollment variations, by increasing speaker discriminability.