The crack propoagation kinetics of binary styrene-(styrene/butadiene)-styrene triblock copolymer blends based on one with symmetrical (LN4) and another with asymmetrical (LN3) molecular architecture is discussed with respect to post-yield crack-tip blunting and stable crack propagation behavior while highlighting the dynamic mechanical properties of the blends. The crack-tip opening displacement (CTOD) rate is revealed to be sensitive to phase behavior, which is in agreement with a transition in phase miscibility in a critical composition range of 40-60 wt.-% of LN3. Analyses of R-curves from CTOD-values reveal that kinetics of crack propagation is controlled by phase behavior, whereas the resistance to stable crack initiation is largely dependent on the composition. Our investigation offers new possibilities to tailor and ontimize the crack resistance (crack propagation stability) of block copolymer blends through the control of phase miscibility and hence, fundamentally, adds a new dimension to the development of novel materials based on toughened nanostructured polymers.