Even if the number of illegal votes is currently negligible, this law will further secure US elections. The only reason Democrats are voting against it is that they're counting on noncitizens casting ballots in their favor. Already, Democrats have been allowing these populations to vote in state elections, and it's only a matter of time before they open the door to federal elections.
The SAVE Act is nothing but blatant voter suppression, as it has the potential to discourage minority voters from registering. In an effort to rile up their base, Republicans are using this bill to stoke racism and xenophobia, while at the same time scaring potential Democratic voters from exercising their constitutional rights.