I have been practicing judo for about 6 months and I've found that I frequently manage to "jam" the big toe on my right foot during randori. Are there strengthening exercises that I can do for my big toe? Should I tape my toes before randori?
You should practice slide-stepping -- with the ball of the foot, arching the toes up. Slowly, slowly, slowly -- quicker -- quicker -- quicker -- full speed.
Until the step becomes so natural that you will not hit your toe in the mat anymore.
And now, practice losing your balance and recovering it with that step.