This problem is so contentious because neither side will admit that the other side might have a point and that they might have legitimate motives. If you are a liberal, consider that currently, one need not validate his or her identity in any way to cast a vote in most places. In Texas, I could collect every voter registration card (delivered by mail on the same day to everyone in the city) and cast one vote for each card I have. The election officials have no way to disqualify my votes because that card is all I need to vote. They are not allowed to ask for any form of verification of my identity and must let me vote. This is obviously ludicrous, but that is the system currently in place. It's additionally absurd since, in many states,
police may demand/require you to produce ID at any time. If citizens have to have ID anyway, then I cannot conceive of any legitimate reason not to require an ID to vote.
If you are a conservative, realize that some people will not get an ID prior to the election even if it's free. Maybe they are out of the loop and don't know they need one, maybe they are old and crotchety, maybe they are lazy and irresponsible. If they show up to vote and can't, then they are denied a fundamental right which everyone should agree is unacceptable. On the other hand, if an ineligible voter shows up and votes, then they have cancelled a vote of an eligible voter which is equally as bad as disenfranchising the eligible voter in the first place.
In the end, we have to draw the line somewhere. Currently, there is no barrier to voter fraud in most places. Since we don't look for it, we can't measure it and we have no way to know whether it's a problem. We have to weigh the possibility of direct disenfranchisement (that is, the case where an eligible voter is not allowed to vote) against the possibility of indirect disenfranchisement (that is, the case where an ineligible voter votes). In my opinion, if some minimal burden is placed on the voter, it will likely tip the scales towards disallowing ineligible votes without unduly burdening eligible voters. The details of how to do this may be debated, but the basic principle seems secure.