Democrats opposed requiring photo ID to vote because they felt it would hurt the minority vote (which is another cornerstone of their party). That's not an unreasonable argument by any stretch of the imagination, think about it, what forms of photo ID do most people have? Driver's license. Some have a college-issued student ID. Minorities are, on the whole, less affluent, thus less likely to own a car or attend college. So I think it's likely that a minority is less likely to have a photo ID (and the means to acquire one). In reality the biggest problem is that minorities are also less educated, thus far more likely to not even know about the rule, and thus get turned away at the polls on election day.
The only way that it would be fair to claim that Democrats want to keep illegals voting because they vote in their favor, is if you found some method of strengthening voter protects that had no unintended consequences which they opposed. Unfortunately, I don't think such a thing exists.
I'm willing to admit that there are probably some Democrats, in the House of Representatives in particular, who live in districts that have a high number of illegal immigrants where voter fraud is common. It would be naive of me to assume that they play no role whatsoever in the American political system, I just think it is absurd to suggest that they have an alarming measure of influence.
As to why there is so much opposition to attempts to strengthen enforcement of illegal immigration laws in the Democratic Party, there's no one reason. Some laws are opposed because they are poorly written and open to abuse. Others because they are too strict on a group of people that Democrats can identify with. And of course others still are laws that a lot of Democrats actually would agree with, but certain subsections of the Democratic Party do not agree with.
There are definitely some Hispanic, legal citizens which are against illegal immigration, but I'd be willing to bet that are the minority. Hispanics trend Democrat and in places like California where they have a lot of political influence, they regularly torpedo attempts to tighten border security or make it harder to be an illegal immigrant living in America, because they identify with people of their heritage.
I don't consider myself a Democratic, I think I'm pretty centrist, but I'm against most attempts at tightening the border by physical means on the grounds of practicality. The border is a long, difficult to patrol stretch of land. I simply don't think it's realistic to build a wall to keep out the illegals, then to repair that wall, to make it impossible to scale or dig beneath, et cetera. Securing the border requires a ridiculous amount of effort.
If I was going to fix the immigration problem, I would do it in a few ways.
1: Universally mandate background and social security checks if you want to be employed in the United States.
Federal employment already requires this, a lot of states do as well, but it needs to be something done everywhere if we want to keep undocumented workers from receiving formal positions. This probably won't help anything when it comes to informal labor such as small construction projects, farms, et cetera, but those aren't the type of positions that a high-school educated American is going for anyway.
2: Strengthen general prosecution of identity theft, and better coordinate the country's identity policing agencies.
This way people who steal social security numbers for the sake of employment will be dealt with, at the same time as we deal with other types of identity theft which are a problem in our society.
3: Start a new work-for-hire program like the very initiative that spawned this problem.