Does visiting a customer = working in another state?

Suppose a person lives in state A, his office and his employer's headquarters are in state A, but from time to time he travels to meet with a customer in state B. Does that constitute performing work in state B, meaning that he might have to file a nonresident return and pay tax to state B for the time he spent there with the customer? For a highly paid person who spends a few days a month with the customer, his salary for those days might exceed the nonresident filing threshold in some states, especially if he is single. Should his employer be withholding state B tax? Does it make any difference whether the employer has nexus in state B? The person never goes to the employer's office in state B (if one exists). All his time in state B is spent at the customer's location.