The existence of synagogue membership fees always seems to be a major shock to the new conversion candidate. Coming from mainstream American society, voluntary donations during a religious service are the expected way for the average person to donate money to a religious organization (whether the candidate was involved in Christianity or not). Now, as a person considering conversion to Judaism in any movement, they hear that synagogue "dues" range from $200-1,000 for a single person! Without paying dues, the person may pay more for synagogue events/tickets and also doesn't get a vote on synagogue policies. Normally, this person gets pretty indignant: "What a ridiculous idea! How could anyone afford that??" In the beginning, the inability to carry money on Shabbat helps placate the candidate a little.
But in short: that's the way things are, and you get used to it. In fact, over time, I've decided this is a much better system.
What are synagogue membership fees? They are a set fee paid to the synagogue for the privilege of being a "member" of the synagogue. They vary widely from synagogue to synagogue and city to city.
What if you can't afford them? Ask for a lower rate. The membership application should even say who should be asked. Yes, this will hurt your ego, but you will be neither the first nor the last person to ask. Every synagogue allows reduced rates for those with low incomes. Still, you will probably be asked to pay more than you really want to. In that case, do the math and see how much you will pay each month or week, and it will probably be a very small number. Considering the amount of money we spend on silly things without thinking, this number will probably be much less.
Worst case scenario: what if you can't pay anything towards membership? They're not going to make you leave. You will still have access to all services and any "regular" classes that don't have a separate fee. You should also have access to any other services that don't require a separate fee. For anything that requires a separate fee, you can most likely use/attend it with only paying the separate fee. For example, you should be able to pay the class or mikvah fee without having to pay a membership fee for the synagogue itself. This also comes into play when traveling, especially for mikvah use. And perhaps most importantly, you can still talk to the rabbi, get rulings, etc. Try not to abuse his time and kindness since membership dues are a significant source for paying his salary.
So what do these membership dues do? In short, they pay the synagogue's bills. The beauty of membership fees for the synagogues is that they are better able to budget their expenses without relying on an uncertain amount of cash in a plate each week.
What do membership dues do for you? First and foremost, you get voting rights within the synagogue community. (Of course, this is the source of shul politics, which is a nasty thing.) You should also get benefits, such as reduced prices for classes, events, mikvah, school for the kids, etc. Each shul will be different. The dues may or may not pay for "High Holyday tickets," which are a similar shock. Basically, the demand for synagogue seats on the High Holydays can be incredibly high because of "Once a Year Jews." Tickets allow the synagogue to guarantee seats to the people who wish to attend, as well as being sure that the number of attendees is below the occupancy limit set by the fire marshall.
Now the $1 million question: Can conversion candidates be synagogue members? It depends on the synagogue. My hunch is that most, if not a significant majority, don't allow it. It's just easier to have a blanket, easy-to-apply policy instead of considering each person on a case-by-case basis. But see the worst case scenario section above; you can still have access to the classes and rabbi, etc. Remember that a synagogue with a policy not allowing pre-converts to be members will not allow a convert whose conversion they don't recognize to join. You will be grouped with pre-converts/non-Jews. For reform converts, most conservative synagogues allow at least some reform converts to be considered Jews for the purposes of the conservative movement. However, even though the national Conservative organization "ruled" in favor of this, every conservative synagogue is able to decide which teshuvot to follow between minority and majority opinions.
All that said, some shuls allow it. Some shuls even allow Christians and other obviously-not-Jewish people to become members! You never know. So ask.
After my conservative conversion, the first thing I did was pay membership dues, and I have never felt so much pride and ownership in turning over a check. I think this is a very tangible way for converts to immediately feel some ownership of the community as a new Jew.