What should I do if I need to do strength training yet I don't want to build muscle?
Any serious strength training is going to change the density and volume of muscle. But some approaches produce a greater obvious volume of muscle - such as you see with competition bodybuilders. And others tend to result in strength gains without as much change in muscle fullness.
It is a matter of how much resistance (weight) you contract the muscles against and how many sets and reps you perform. As a basic rule, lifting heavier weights for fewer reps (maximum 6–8 reps, target volume 1–6 reps) tends to create more strength but less muscle size while doing more reps the way a bodybuilders does (3 to 4 sets of 3 to 4 exercises, 10–12 reps for upper body and 12 to 16 reps for legs) produces less absolute strength but more muscle fullness and shape.
Train like a weightlifter for more strength, like a bodybuilder for shape and muscularity. Or use both styles at different times for maximum overall results.
Of course genetics being such a determining factor that the above strategies are only approximations. You may find you are getting more of one result compared to another because of your specific genetic nature. But this is a good rule of thumb to use in planning your exercise program.