Yeah, doesn't seem that the proposed bill offers meaningful improvements. The article says

Under current New Hampshire law, voters are asked to provide proof of identity and age (usually a driver’s license), proof that they live where they want to vote, and proof of citizenship (either a birth certificate or a passport) in order to be able to register and vote.

It's only if a person doesn't have a proof of citizenship, they can sign a sworn affidavit to register or vote, which will be checked by attorney general office after elections.

This looks to me as a solid system already. One cannot vote by impersonating someone else, one cannot vote if you are not living in the district, one can only lie about citizenship with a significant risk of that lie to be exposed after elections with all legal consequences.

Unless you belive that damn liberals moving buses of illegals with forged IDs to steal the elections, there is nothing much to be worried about.