When I first started playing, I was having a real problem with constantly hitting the wrong string, or sometimes just flat missing the icho-no-ito altogether and only hitting the skin/head. I finally figured out that I was just holding the shamisen wrong, supporting the sao with my hand, rather than balancing the shamisen as others have pointed out. Moving quickly from note to note would cause the whole body of the shamisen to jerk, misdirecting my bachi strike.
When discussing this form problem with my sensei, the inestimable Mike Penny, he told me that when he learned to play in Japan, he wasn’t allowed to touch the shamisen with his left hand for some really long period of time (I can’t remember how long exactly, but it was multiple months). He could only balance the shamisen with his right arm and strike the strings with the bachi. Crazy! But Mike’s form is awesome, so I guess it paid off in the end…