Looking at your wish list, you're not abstracting anything away at all. You're just adding a needless layer.

open a DBI connection given a hostname, database, username and password;

DBI->connect

the user (if in an interactive context) for any of the four things above if missing;

Bad idea. Data storage and user interaction are unrelated tasks.

supply a method for preparing a query;

$dbh->prepare

supply a method for executing a prepared query;

$sth->execute

allow the user to just call DBModule::run_query("select ... from ... where ...") and expect the module to properly prepare/execute

If $dbh->prepare + $sth->execute is really a hardship,

sub do_select {
   my($dbh, $statement, $attr, @bind_values) = @_;
   my $sth = $dbh->prepare($statement, $attr) or return undef;
   $sth->execute(@bind_values) or return undef;
   return $sth;
}
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while helpfully handling / trapping Sybase error codes

RaiseError => 1

supply a method to reformat the result set into various hashes/arrays

$dbh->selectall_*

supply a method to release a result set (equiv to $sth->finish()

$sth->finish, though rarely needed.