The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. Applications include a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single columns and single rows, we give a new interpretation of Lascoux polynomials ($K$-analogues of Demazure characters) by constructing a $K$-theoretic analogue of crystals with an appropriate analogue of a Demazure crystal. (Joint work with Cara Monical and Travis Scrimshaw.)