Yes, they can. California does this today. California has a border patrol that will throw away any contraband you attempt to smuggle in to the state of California. Border stations exist between Mexico and California, Nevada and California, Arizona and California, and Oregon and California. For example, if California discovers you have a ferret, they will fine you or put you in jail. If you attempt to bring a ferret into California via the interstate highway, and go through a california border station, they will likely confiscate and euthanize the ferret. This is not just for ferrets but all sorts of pests, plants, etc. California keeps a tight border. Last time I crossed the border from Oregon, we were stopped, and questioned, and our car inspected with a mirror underneath, at the border station.
The Constitution guarantees that you -- an American citizen or legal immigrant or tourist -- can enter and leave the state at will -- not your stuff.