The original, unamended Constitution does not say who can or cannot vote. The Constitution specifically leaves it up to the states:
Article I. Section 2. The House of Representatives shall be composed of Members chosen every second Year by the People of the several States, and the Electors in each State shall have the Qualifications requisite for Electors of the most numerous Branch of the State Legislature.
Senators were chosen by state legislatures. The president, of course, by Electors.
The subsequent amendments for race, color and prior servitude (15th), gender (19th), poll tax (24th) and age (26th) simply told the states that they could not deny the vote for those reasons.