If you are driving, then you always need to carry a driver's license while operating a vehicle. Therefore, if you are moving between states by driving, you will need to have your license. However, the driver's license is not necessary to the right to travel.
357 U. S. 117-130. (a) The right to travel is a part of the "liberty" of which a citizen cannot be deprived without due process of law under the Fifth Amendment. Pp. 357 U. S. 125-127.
However, traveling does not equate to driving. In the Constitution's view, travel refers to moving freely between states, not the unlicensed operation of a vehicle. This point was strongly reinforced in Saenz v. Roe (1999), rejecting the belief in an absolute, unrestricted freedom.
Is it constitutional for the government to require a license to drive? There's nothing in the US Constitution giving the Federal government any right to license drivers.
Article IV, Section 2, Clause 1: The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.
A: If you're going to operate a motor vehicle on public roads, you need a license, insurance and vehicle registration. You have a right to travel, but doing it in a motor vehicle on public roads is a privilege and the government regulates that.
It's a little more complex than that. The Supreme Court has ruled that there is a fundamental right to travel between the states, and you do not need a drivers license to do so.
There is no explicit or enumerated right to travel in the US Constitution. It is, however, considered to be a Ninth Amendment right, similar to the right to vote and the former right to abortion.