Voting in the 1788 Presidential election seems to have been carried out by public ballot, although some states continued to use an oral ballot until the late nineteenth century (e.g. Kentucky, which continued to use an oral ballot until 1891).
Secret ballots, also known as "Australian Ballots" were introduced to the United States in the nineteenth century, but were only extensively adopted after the 1884 Presidential election.
Secret ballots system were introduced in the United Kingdom by the Ballot Act of 1872. Prior to that, a public ballot system had been in use with the votes cast by individuals recorded in Poll Books. These poll books had been introduced by an Act of Parliament in 1696
"to stop electoral fraud and were the responsibility of the local Sheriffs".
- Poll books - University of Reading Special Collections
There is an interesting article, Public Voting: Before the Secret Ballot, from The Institute for Advanced Technology in the Humanities at the University of Virginia. It discusses the various methods of voting used in the United States before the secret ballot was adopted in the later nineteenth century.