If I were Emperor of the country, I'd have the following system in place:
#1 Any eligible person can register prior to the election but not within 30 days of the election for which they are registering. Earlier registration is permitted based on age and residency requirements being met by the day of the ballot casting.
#2 At the time of registration the individual must present proof of identity and eligibility to vote (i.e. citizenship).
#3 Between registration and being eligible to vote the Registrar will ensure the individual is (a) a citizen, and (b) that their right to vote has not be removed by legal action (or has been restored as the case may be).
#4 Permissible ID's must be issued under State authority and will show "Vote: Yes" or "Vote: No". (No Illegal Alien should, IMHO, eligible for a state issued ID. Legal residence are also eligible to drive or have need for a non-driving ID - therefore the ID should clearly indicate voter eligibility.)
#5 The State will issue both Driver Licenses and non-Driving ID at their normal cost schedule. If an ID is needed for another purpose and has a cost, then normal cost factors apply. If the individual is indigent and cannot afford one, then a non-Driving ID will be issued that will clearly indicate (through text, color, etc.) "This ID is election ballot casting only and is not valid for any other purpose." The individual is then required to sign a statement of understanding regarding the use and law for that ID. A companion law would then be passed making the use (or acceptance) of the Voter Only ID as misdemeanor offense punishable by a $500 fine for the User and the Individual accepting such an ID for other than voting purposes.
#6 Most states already use a "Motor/Voter" type of system. States already have to ability to access driving records from other states through an exchange process. Added to this would be that when an individual moves and they register in a new location, the registration at their previous location is updated to "Invalid".
#7 Each State will establish formal procedures to deal with the (extremely rare) cases of a person born a citizen yet who is unable to obtain a valid birth certificate (as in one does not exist - not to be confused with someone that waits until the last minute and doesn't have one on hand).
#8 To alleviate the cost of obtaining a birth certificate and individual can order one of two types of certificates: (a) general use, or (b) voter only. If the individual orders a general use birth certificate they can use it for whatever purposes they desire (drivers license, employment, passports, etc.) with no restrictions. If used for voter registration the registrar examines (and copies if needed) the birth certificate and the original is returned to the individual. However a "Voter Only" birth certificate can only be used for registration purposes. Again a companion law would fine individuals using and/or accepting such birth certificates for other purposes. Once the "Voter Only" certificate is presented to the registrar, the eligibility is validated and the registrar retains the certificate for destruction. States would accept reciprocity (just as they do now with Drivers Licenses and non-Driving ID's. If an individual has passed the citizenship check and has an ID from State A showing eligibility to vote, then there is no need to get another Voter Only birth certificate, it's use in the previous state to generate the ID is recognized.
That's what I can think of off the top of my head.