Yes, absolutely. The 14th Amendment (1868 ) very specifically addresses this very point. In theory, we are all to be treated equally under the law and benefit from the equal protection of the laws. This is called the Equal Protection Clause of the 14 Amendment.
Due process of the law is also included in the 14th Amendment, hence it is also called the Due Process Clause of the 14th Amendment. The doctrine of due process protects citizens from arbitrary governmental actions. The same laws and legal procedures are the same for all citizens regardless of circumstances or race or ability to hire legal counsel. Due process also specifies procedural issues that the state must follow when initiating a criminal action against a criminal defendant. Again, these procedures are the same for all citizens... the very richest and the very poorest.
One reason the state appoints counsel for indigent defendants is because of the Equal Protection Clause. All people, regardless of ability to pay, have a right to competent legal representation.